- 6.6.1 Find the parametric representations of a cylinder, a cone, and a sphere.
- 6.6.2 Describe the surface integral of a scalar-valued function over a parametric surface.
- 6.6.3 Use a surface integral to calculate the area of a given surface.
- 6.6.4 Explain the meaning of an oriented surface, giving an example.
- 6.6.5 Describe the surface integral of a vector field.
- 6.6.6 Use surface integrals to solve applied problems.
We have seen that a line integral is an integral over a path in a plane or in space. However, if we wish to integrate over a surface (a two-dimensional object) rather than a path (a one-dimensional object) in space, then we need a new kind of integral that can handle integration over objects in higher dimensions. We can extend the concept of a line integral to a surface integral to allow us to perform this integration.
Surface integrals are important for the same reasons that line integrals are important. They have many applications to physics and engineering, and they allow us to develop higher dimensional versions of the Fundamental Theorem of Calculus. In particular, surface integrals allow us to generalize Green’s theorem to higher dimensions, and they appear in some important theorems we discuss in later sections.
A surface integral is similar to a line integral, except the integration is done over a surface rather than a path. In this sense, surface integrals expand on our study of line integrals. Just as with line integrals, there are two kinds of surface integrals: a surface integral of a scalar-valued function and a surface integral of a vector field.
However, before we can integrate over a surface, we need to consider the surface itself. Recall that to calculate a scalar or vector line integral over curve C, we first need to parameterize C. In a similar way, to calculate a surface integral over surface S, we need to parameterize S. That is, we need a working concept of a parameterized surface (or a parametric surface), in the same way that we already have a concept of a parameterized curve.
A parameterized surface is given by a description of the form
Notice that this parameterization involves two parameters, u and v, because a surface is two-dimensional, and therefore two variables are needed to trace out the surface. The parameters u and v vary over a region called the parameter domain, or parameter space—the set of points in the uv-plane that can be substituted into r. Each choice of u and v in the parameter domain gives a point on the surface, just as each choice of a parameter t gives a point on a parameterized curve. The entire surface is created by making all possible choices of u and v over the parameter domain.
Given a parameterization of surface the parameter domain of the parameterization is the set of points in the uv-plane that can be substituted into r.
Parameterizing a Cylinder
Describe surface S parameterized by
Notice that if we change the parameter domain, we could get a different surface. For example, if we restricted the domain to then the surface would be a half-cylinder of height 6.
Describe the surface with parameterization
It follows from Example 6.58 that we can parameterize all cylinders of the form If S is a cylinder given by equation then a parameterization of S is
We can also find different types of surfaces given their parameterization, or we can find a parameterization when we are given a surface.
Describing a Surface
Describe surface S parameterized by
Describe the surface parameterized by
Finding a Parameterization
Give a parameterization of the cone lying on or above the plane
Give a parameterization for the portion of cone lying in the first octant.
We have discussed parameterizations of various surfaces, but two important types of surfaces need a separate discussion: spheres and graphs of two-variable functions. To parameterize a sphere, it is easiest to use spherical coordinates. The sphere of radius centered at the origin is given by the parameterization
The idea of this parameterization is that as sweeps downward from the positive z-axis, a circle of radius is traced out by letting run from 0 to To see this, let be fixed. Then
This results in the desired circle (Figure 6.61).
Finally, to parameterize the graph of a two-variable function, we first let be a function of two variables. The simplest parameterization of the graph of is where x and y vary over the domain of (Figure 6.62). For example, the graph of can be parameterized by where the parameters x and y vary over the domain of If we only care about a piece of the graph of —say, the piece of the graph over rectangle —then we can restrict the parameter domain to give this piece of the surface:
Similarly, if S is a surface given by equation or equation then a parameterization of S is
or respectively. For example, the graph of paraboloid can be parameterized by Notice that we do not need to vary over the entire domain of y because x and z are squared.
Let’s now generalize the notions of smoothness and regularity to a parametric surface. Recall that curve parameterization is regular if for all t in For a curve, this condition ensures that the image of r really is a curve, and not just a point. For example, consider curve parameterization The image of this parameterization is simply point which is not a curve. Notice also that The fact that the derivative is zero indicates we are not actually looking at a curve.
Analogously, we would like a notion of regularity for surfaces so that a surface parameterization really does trace out a surface. To motivate the definition of regularity of a surface parameterization, consider parameterization
Although this parameterization appears to be the parameterization of a surface, notice that the image is actually a line (Figure 6.63). How could we avoid parameterizations such as this? Parameterizations that do not give an actual surface? Notice that and and the corresponding cross product is zero. The analog of the condition is that is not zero for point in the parameter domain, which is a regular parameterization.
Parameterization is a regular parameterization if is not zero for point in the parameter domain.
If parameterization r is regular, then the image of r is a two-dimensional object, as a surface should be. Throughout this chapter, parameterizations are assumed to be regular.
Recall that curve parameterization is smooth if is continuous and for all t in Informally, a curve parameterization is smooth if the resulting curve has no sharp corners. The definition of a smooth surface parameterization is similar. Informally, a surface parameterization is smooth if the resulting surface has no sharp corners.
A surface parameterization is smooth if vector is not zero for any choice of u and v in the parameter domain.
A surface may also be piecewise smooth if it has smooth faces but also has locations where the directional derivatives do not exist.
Is the surface parameterization smooth?
Surface Area of a Parametric Surface
Our goal is to define a surface integral, and as a first step we have examined how to parameterize a surface. The second step is to define the surface area of a parametric surface. The notation needed to develop this definition is used throughout the rest of this chapter.
Let S be a surface with parameterization over some parameter domain D. We assume here and throughout that the surface parameterization is continuously differentiable—meaning, each component function has continuous partial derivatives. Assume for the sake of simplicity that D is a rectangle (although the following material can be extended to handle nonrectangular parameter domains). Divide rectangle D into subrectangles with horizontal width and vertical length Suppose that i ranges from 1 to m and j ranges from 1 to n so that D is subdivided into mn rectangles. This division of D into subrectangles gives a corresponding division of surface S into pieces Choose point in each piece Point corresponds to point in the parameter domain.
Note that we can form a grid with lines that are parallel to the u-axis and the v-axis in the uv-plane. These grid lines correspond to a set of grid curves on surface S that is parameterized by Without loss of generality, we assume that is located at the corner of two grid curves, as in Figure 6.65. If we think of r as a mapping from the uv-plane to the grid curves are the image of the grid lines under r. To be precise, consider the grid lines that go through point One line is given by the other is given by In the first grid line, the horizontal component is held constant, yielding a vertical line through In the second grid line, the vertical component is held constant, yielding a horizontal line through The corresponding grid curves are and and these curves intersect at point
Now consider the vectors that are tangent to these grid curves. For grid curve the tangent vector at is
For grid curve the tangent vector at is
If vector exists and is not zero, then the tangent plane at exists (Figure 6.66). If piece is small enough, then the tangent plane at point is a good approximation of piece
The tangent plane at contains vectors and and therefore the parallelogram spanned by and is in the tangent plane. Since the original rectangle in the uv-plane corresponding to has width and length the parallelogram that we use to approximate is the parallelogram spanned by and In other words, we scale the tangent vectors by the constants and to match the scale of the original division of rectangles in the parameter domain. Therefore, the area of the parallelogram used to approximate the area of is
Varying point over all pieces and the previous approximation leads to the following definition of surface area of a parametric surface (Figure 6.67).
Let with parameter domain D be a smooth parameterization of surface S. Furthermore, assume that S is traced out only once as varies over D. The surface area of S is
Calculating Surface Area
Calculate the lateral surface area (the area of the “side,” not including the base) of the right circular cone with height h and radius r.
The surface area of a right circular cone with radius r and height h is usually given as The reason for this is that the circular base is included as part of the cone, and therefore the area of the base is added to the lateral surface area that we found.
Find the surface area of the surface with parameterization
Calculating Surface Area
Show that the surface area of the sphere is
Show that the surface area of cylinder is Notice that this cylinder does not include the top and bottom circles.
In addition to parameterizing surfaces given by equations or standard geometric shapes such as cones and spheres, we can also parameterize surfaces of revolution. Therefore, we can calculate the surface area of a surface of revolution by using the same techniques. Let be a positive single-variable function on the domain and let S be the surface obtained by rotating about the x-axis (Figure 6.69). Let be the angle of rotation. Then, S can be parameterized with parameters x and by
Surface Integral of a Scalar-Valued Function
Now that we can parameterize surfaces and we can calculate their surface areas, we are able to define surface integrals. First, let’s look at the surface integral of a scalar-valued function. Informally, the surface integral of a scalar-valued function is an analog of a scalar line integral in one higher dimension. The domain of integration of a scalar line integral is a parameterized curve (a one-dimensional object); the domain of integration of a scalar surface integral is a parameterized surface (a two-dimensional object). Therefore, the definition of a surface integral follows the definition of a line integral quite closely. For scalar line integrals, we chopped the domain curve into tiny pieces, chose a point in each piece, computed the function at that point, and took a limit of the corresponding Riemann sum. For scalar surface integrals, we chop the domain region (no longer a curve) into tiny pieces and proceed in the same fashion.
Let S be a piecewise smooth surface with parameterization with parameter domain D and let be a function with a domain that contains S. For now, assume the parameter domain D is a rectangle, but we can extend the basic logic of how we proceed to any parameter domain (the choice of a rectangle is simply to make the notation more manageable). Divide rectangle D into subrectangles with horizontal width and vertical length Suppose that i ranges from 1 to m and j ranges from 1 to n so that D is subdivided into mn rectangles. This division of D into subrectangles gives a corresponding division of S into pieces Choose point in each piece evaluate at , and multiply by area to form the Riemann sum
To define a surface integral of a scalar-valued function, we let the areas of the pieces of S shrink to zero by taking a limit.
The surface integral of a scalar-valued function of over a piecewise smooth surface S is
Again, notice the similarities between this definition and the definition of a scalar line integral. In the definition of a line integral we chop a curve into pieces, evaluate a function at a point in each piece, and let the length of the pieces shrink to zero by taking the limit of the corresponding Riemann sum. In the definition of a surface integral, we chop a surface into pieces, evaluate a function at a point in each piece, and let the area of the pieces shrink to zero by taking the limit of the corresponding Riemann sum. Thus, a surface integral is similar to a line integral but in one higher dimension.
The definition of a scalar line integral can be extended to parameter domains that are not rectangles by using the same logic used earlier. The basic idea is to chop the parameter domain into small pieces, choose a sample point in each piece, and so on. The exact shape of each piece in the sample domain becomes irrelevant as the areas of the pieces shrink to zero.
Scalar surface integrals are difficult to compute from the definition, just as scalar line integrals are. To develop a method that makes surface integrals easier to compute, we approximate surface areas with small pieces of a tangent plane, just as we did in the previous subsection. Recall the definition of vectors and
From the material we have already studied, we know that
This approximation becomes arbitrarily close to as we increase the number of pieces by letting m and n go to infinity. Therefore, we have the following equation to calculate scalar surface integrals:
In this case, vector is perpendicular to the surface, whereas vector is tangent to the curve.
Calculating a Surface Integral
Calculate surface integral where is the surface with parameterization for and
Calculate where S is the surface with parameterization
Calculating the Surface Integral of a Piece of a Sphere
Calculate surface integral where and S is the surface that consists of the piece of sphere that lies on or above plane and the disk that is enclosed by intersection plane and the given sphere (Figure 6.72).
In this example we broke a surface integral over a piecewise surface into the addition of surface integrals over smooth subsurfaces. There were only two smooth subsurfaces in this example, but this technique extends to finitely many smooth subsurfaces.
Calculate line integral where S is cylinder including the circular top and bottom.
Scalar surface integrals have several real-world applications. Recall that scalar line integrals can be used to compute the mass of a wire given its density function. In a similar fashion, we can use scalar surface integrals to compute the mass of a sheet given its density function. If a thin sheet of metal has the shape of surface S and the density of the sheet at point is then mass m of the sheet is
Calculating the Mass of a Sheet
A flat sheet of metal has the shape of surface that lies above rectangle and If the density of the sheet is given by what is the mass of the sheet?
A piece of metal has a shape that is modeled by paraboloid and the density of the metal is given by Find the mass of the piece of metal.
Orientation of a Surface
Recall that when we defined a scalar line integral, we did not need to worry about an orientation of the curve of integration. The same was true for scalar surface integrals: we did not need to worry about an “orientation” of the surface of integration.
On the other hand, when we defined vector line integrals, the curve of integration needed an orientation. That is, we needed the notion of an oriented curve to define a vector line integral without ambiguity. Similarly, when we define a surface integral of a vector field, we need the notion of an oriented surface. An oriented surface is given an “upward” or “downward” orientation or, in the case of surfaces such as a sphere or cylinder, an “outward” or “inward” orientation.
Let S be a smooth surface. For any point on S, we can identify two unit normal vectors and If it is possible to choose a unit normal vector N at every point on S so that N varies continuously over S, then S is “orientable.” Such a choice of unit normal vector at each point gives the orientation of a surface S. If you think of the normal field as describing water flow, then the side of the surface that water flows toward is the “negative” side and the side of the surface at which the water flows away is the “positive” side. Informally, a choice of orientation gives S an “outer” side and an “inner” side (or an “upward” side and a “downward” side), just as a choice of orientation of a curve gives the curve “forward” and “backward” directions.
Closed surfaces such as spheres are orientable: if we choose the outward normal vector at each point on the surface of the sphere, then the unit normal vectors vary continuously. This is called the positive orientation of the closed surface (Figure 6.74). We also could choose the inward normal vector at each point to give an “inward” orientation, which is the negative orientation of the surface.
A portion of the graph of any smooth function is also orientable. If we choose the unit normal vector that points “above” the surface at each point, then the unit normal vectors vary continuously over the surface. We could also choose the unit normal vector that points “below” the surface at each point. To get such an orientation, we parameterize the graph of in the standard way: where x and y vary over the domain of Then, and and therefore the cross product (which is normal to the surface at any point on the surface) is Since the z component of this vector is one, the corresponding unit normal vector points “upward,” and the upward side of the surface is chosen to be the “positive” side.
Let S be a smooth orientable surface with parameterization For each point on the surface, vectors and lie in the tangent plane at that point. Vector is normal to the tangent plane at and is therefore normal to S at that point. Therefore, the choice of unit normal vector
gives an orientation of surface S.
Choosing an Orientation
Give an orientation of cylinder
Give the “upward” orientation of the graph of
Since every curve has a “forward” and “backward” direction (or, in the case of a closed curve, a clockwise and counterclockwise direction), it is possible to give an orientation to any curve. Hence, it is possible to think of every curve as an oriented curve. This is not the case with surfaces, however. Some surfaces cannot be oriented; such surfaces are called nonorientable. Essentially, a surface can be oriented if the surface has an “inner” side and an “outer” side, or an “upward” side and a “downward” side. Some surfaces are twisted in such a fashion that there is no well-defined notion of an “inner” or “outer” side.
The classic example of a nonorientable surface is the Möbius strip. To create a Möbius strip, take a rectangular strip of paper, give the piece of paper a half-twist, and the glue the ends together (Figure 6.76). Because of the half-twist in the strip, the surface has no “outer” side or “inner” side. If you imagine placing a normal vector at a point on the strip and having the vector travel all the way around the band, then (because of the half-twist) the vector points in the opposite direction when it gets back to its original position. Therefore, the strip really only has one side.
Since some surfaces are nonorientable, it is not possible to define a vector surface integral on all piecewise smooth surfaces. This is in contrast to vector line integrals, which can be defined on any piecewise smooth curve.
Surface Integral of a Vector Field
With the idea of orientable surfaces in place, we are now ready to define a surface integral of a vector field. The definition is analogous to the definition of the flux of a vector field along a plane curve. Recall that if F is a two-dimensional vector field and C is a plane curve, then the definition of the flux of F along C involved chopping C into small pieces, choosing a point inside each piece, and calculating at the point (where N is the unit normal vector at the point). The definition of a surface integral of a vector field proceeds in the same fashion, except now we chop surface S into small pieces, choose a point in the small (two-dimensional) piece, and calculate at the point.
To place this definition in a real-world setting, let S be an oriented surface with unit normal vector N. Let v be a velocity field of a fluid flowing through S, and suppose the fluid has density Imagine the fluid flows through S, but S is completely permeable so that it does not impede the fluid flow (Figure 6.77). The mass flux of the fluid is the rate of mass flow per unit area. The mass flux is measured in mass per unit time per unit area. How could we calculate the mass flux of the fluid across S?
The rate of flow, measured in mass per unit time per unit area, is To calculate the mass flux across S, chop S into small pieces If is small enough, then it can be approximated by a tangent plane at some point P in Therefore, the unit normal vector at P can be used to approximate across the entire piece because the normal vector to a plane does not change as we move across the plane. The component of the vector at P in the direction of N is at P. Since is small, the dot product changes very little as we vary across and therefore can be taken as approximately constant across To approximate the mass of fluid per unit time flowing across (and not just locally at point P), we need to multiply by the area of Therefore, the mass of fluid per unit time flowing across in the direction of N can be approximated by where N, and v are all evaluated at P (Figure 6.78). This is analogous to the flux of two-dimensional vector field F across plane curve C, in which we approximated flux across a small piece of C with the expression To approximate the mass flux across S, form the sum As pieces get smaller, the sum gets arbitrarily close to the mass flux. Therefore, the mass flux is
This is a surface integral of a vector field. Letting the vector field be an arbitrary vector field F leads to the following definition.
Let F be a continuous vector field with a domain that contains oriented surface S with unit normal vector N. The surface integral of F over S is
Notice the parallel between this definition and the definition of vector line integral A surface integral of a vector field is defined in a similar way to a flux line integral across a curve, except the domain of integration is a surface (a two-dimensional object) rather than a curve (a one-dimensional object). Integral is called the flux of F across S, just as integral is the flux of F across curve C. A surface integral over a vector field is also called a flux integral.
Just as with vector line integrals, surface integral is easier to compute after surface S has been parameterized. Let be a parameterization of S with parameter domain D. Then, the unit normal vector is given by and, from Equation 6.20, we have
Therefore, to compute a surface integral over a vector field we can use the equation
Calculating a Surface Integral
Calculate the surface integral where and S is the surface with parameterization
Calculate surface integral where and S is the portion of the unit sphere in the first octant with outward orientation.
Calculating Mass Flow Rate
Let represent a velocity field (with units of meters per second) of a fluid with constant density 80 kg/m3. Let S be hemisphere with such that S is oriented outward. Find the mass flow rate of the fluid across S.
Let m/sec represent a velocity field of a fluid with constant density 100 kg/m3. Let S be the half-cylinder oriented outward. Calculate the mass flux of the fluid across S.
In Example 6.70, we computed the mass flux, which is the rate of mass flow per unit area. If we want to find the flow rate (measured in volume per time) instead, we can use flux integral which leaves out the density. Since the flow rate of a fluid is measured in volume per unit time, flow rate does not take mass into account. Therefore, we have the following characterization of the flow rate of a fluid with velocity v across a surface S:
To compute the flow rate of the fluid in Example 6.68, we simply remove the density constant, which gives a flow rate of
Both mass flux and flow rate are important in physics and engineering. Mass flux measures how much mass is flowing across a surface; flow rate measures how much volume of fluid is flowing across a surface.
In addition to modeling fluid flow, surface integrals can be used to model heat flow. Suppose that the temperature at point in an object is Then the heat flow is a vector field proportional to the negative temperature gradient in the object. To be precise, the heat flow is defined as vector field where the constant k is the thermal conductivity of the substance from which the object is made (this constant is determined experimentally). The rate of heat flow across surface S in the object is given by the flux integral
Calculating Heat Flow
A cast-iron solid cylinder is given by inequalities The temperature at point in a region containing the cylinder is Given that the thermal conductivity of cast iron is 55, find the heat flow across the boundary of the solid if this boundary is oriented outward.
A cast-iron solid ball is given by inequality The temperature at a point in a region containing the ball is Find the heat flow across the boundary of the solid if this boundary is oriented outward.
Section 6.6 Exercises
For the following exercises, determine whether the statements are true or false.
If surface S is given by then
If surface S is given by then
Surface is the same as surface for
Given the standard parameterization of a sphere, normal vectors are outward normal vectors.
For the following exercises, find parametric descriptions for the following surfaces.
The frustum of cone
The portion of cylinder in the first octant, for
A cone with base radius r and height h, where r and h are positive constants
For the following exercises, use a computer algebra system to approximate the area of the following surfaces using a parametric description of the surface.
[T] Half cylinder
[T] Plane above square
For the following exercises, let S be the hemisphere with and evaluate each surface integral.
For the following exercises, evaluate for vector field F, where N is an upward pointing normal vector to surface S.
and S is that part of plane that lies above unit square
and S is hemisphere
and S is the portion of plane that lies inside cylinder
For the following exercises, approximate the mass of the homogeneous lamina that has the shape of given surface S. Round to four decimal places.
[T] S is surface
[T] S is surface
[T] S is surface
Evaluate where S is the surface of cube Assume an outward pointing normal.
Evaluate surface integral where and S is the portion of plane that lies over unit square R:
Evaluate where S is the surface defined parametrically by for
[T] Evaluate where S is the surface defined by
[T] Evaluate where is the surface defined by
Evaluate where S is the surface bounded above hemisphere and below by plane
Evaluate where S is the portion of plane that lies inside cylinder
[T] Evaluate where S is the portion of cone that lies between planes and
[T] Evaluate where S is the portion of cylinder that lies in the first octant between planes and
[T] Evaluate where S is the part of the graph of in the first octant between the xz-plane and plane
Evaluate if S is the part of plane that lies over the triangular region in the xy-plane with vertices (0, 0, 0), (1, 0, 0), and (0, 2, 0).
Find the mass of a lamina of density in the shape of hemisphere
Compute where and N is an outward normal vector S, where S is the union of two squares and
Compute where and N is an upward pointing normal vector S, where S is the triangular region cut off from plane by the positive coordinate axes.
Compute where and N is an outward normal vector S, where S is the surface of sphere
Compute where and N is an outward normal vector S, where S is the surface of the five faces of the unit cube missing
For the following exercises, express the surface integral as an iterated double integral by using a projection on S on the yz-plane.
S is the first-octant portion of plane
S is the portion of the graph of bounded by the coordinate planes and plane
For the following exercises, express the surface integral as an iterated double integral by using a projection on S on the xz-plane
S is the first-octant portion of plane
S is the portion of the graph of bounded by the coordinate planes and plane
Evaluate surface integral where S is the first-octant part of plane where is a positive constant.
Evaluate surface integral where S is hemisphere
Evaluate surface integral where S is surface
Evaluate surface integral where S is the part of plane that lies above rectangle
Evaluate surface integral where S is plane that lies in the first octant.
Evaluate surface integral where S is the part of plane that lies inside cylinder
For the following exercises, use geometric reasoning to evaluate the given surface integrals.
where S is surface
where S is surface oriented with unit normal vectors pointing outward
where S is disc on plane oriented with unit normal vectors pointing upward
A lamina has the shape of a portion of sphere that lies within cone Let S be the spherical shell centered at the origin with radius a, and let C be the right circular cone with a vertex at the origin and an axis of symmetry that coincides with the z-axis. Determine the mass of the lamina if
A lamina has the shape of a portion of sphere that lies within cone Let S be the spherical shell centered at the origin with radius a, and let C be the right circular cone with a vertex at the origin and an axis of symmetry that coincides with the z-axis. Suppose the vertex angle of the cone is Determine the mass of that portion of the shape enclosed in the intersection of S and C. Assume
A paper cup has the shape of an inverted right circular cone of height 6 in. and radius of top 3 in. If the cup is full of water weighing find the total force exerted by the water on the inside surface of the cup.
For the following exercises, the heat flow vector field for conducting objects i is the temperature in the object and is a constant that depends on the material. Find the outward flux of F across the following surfaces S for the given temperature distributions and assume