7.2.1 Determine derivatives and equations of tangents for parametric curves.
7.2.2 Find the area under a parametric curve.
7.2.3 Use the equation for arc length of a parametric curve.
7.2.4 Apply the formula for surface area to a volume generated by a parametric curve.
Now that we have introduced the concept of a parameterized curve, our next step is to learn how to work with this concept in the context of calculus. For example, if we know a parameterization of a given curve, is it possible to calculate the slope of a tangent line to the curve? How about the arc length of the curve? Or the area under the curve?
Another scenario: Suppose we would like to represent the location of a baseball after the ball leaves a pitcher’s hand. If the position of the baseball is represented by the plane curve then we should be able to use calculus to find the speed of the ball at any given time. Furthermore, we should be able to calculate just how far that ball has traveled as a function of time.
Derivatives of Parametric Equations
We start by asking how to calculate the slope of a line tangent to a parametric curve at a point. Consider the plane curve defined by the parametric equations
The graph of this curve appears in Figure 7.16. It is a line segment starting at and ending at
Figure 7.16Graph of the line segment described by the given parametric equations.
We can eliminate the parameter by first solving the equation for t:
Substituting this into we obtain
The slope of this line is given by Next we calculate and This gives and Notice that This is no coincidence, as outlined in the following theorem.
Theorem 7.1
Derivative of Parametric Equations
Consider the plane curve defined by the parametric equations and Suppose that and exist, and assume that Then the derivative is given by
7.1
Proof
This theorem can be proven using the Chain Rule. In particular, assume that the parameter t can be eliminated, yielding a differentiable function Then Differentiating both sides of this equation using the Chain Rule yields
so
But which proves the theorem.
□
Equation 7.1 can be used to calculate derivatives of plane curves, as well as critical points. Recall that a critical point of a differentiable function is any point such that either or does not exist. Equation 7.1 gives a formula for the slope of a tangent line to a curve defined parametrically regardless of whether the curve can be described by a function or not.
Example 7.4
Finding the Derivative of a Parametric Curve
Calculate the derivative for each of the following parametrically defined plane curves, and locate any critical points on their respective graphs.
Solution
To apply Equation 7.1, first calculate and
Next substitute these into the equation:
This derivative is undefined when Calculating and gives and which corresponds to the point on the graph. The graph of this curve is a parabola opening to the right, and the point is its vertex as shown.
Figure 7.17Graph of the parabola described by parametric equations in part a.
To apply Equation 7.1, first calculate and
Next substitute these into the equation:
This derivative is zero when When we have
which corresponds to the point on the graph. When we have
which corresponds to the point on the graph. The point is a relative minimum and the point is a relative maximum, as seen in the following graph.
Figure 7.18Graph of the curve described by parametric equations in part b.
To apply Equation 7.1, first calculate and
Next substitute these into the equation:
This derivative is zero when and is undefined when This gives as critical points for t. Substituting each of these into and we obtain
0
5
0
0
5
−5
0
0
−5
5
0
These points correspond to the sides, top, and bottom of the circle that is represented by the parametric equations (Figure 7.19). On the left and right edges of the circle, the derivative is undefined, and on the top and bottom, the derivative equals zero.
Figure 7.19Graph of the curve described by parametric equations in part c.
Checkpoint 7.4
Calculate the derivative for the plane curve defined by the equations
and locate any critical points on its graph.
Example 7.5
Finding a Tangent Line
Find the equation of the tangent line to the curve defined by the equations
Solution
First find the slope of the tangent line using Equation 7.1, which means calculating and
Next substitute these into the equation:
When so this is the slope of the tangent line. Calculating and gives
which corresponds to the point on the graph (Figure 7.20). Now use the point-slope form of the equation of a line to find the equation of the tangent line:
Figure 7.20Tangent line to the parabola described by the given parametric equations when
Checkpoint 7.5
Find the equation of the tangent line to the curve defined by the equations
Second-Order Derivatives
Our next goal is to see how to take the second derivative of a function defined parametrically. The second derivative of a function is defined to be the derivative of the first derivative; that is,
Since we can replace the on both sides of this equation with This gives us
7.2
If we know as a function of t, then this formula is straightforward to apply.
Example 7.6
Finding a Second Derivative
Calculate the second derivative for the plane curve defined by the parametric equations
Calculate the second derivative for the plane curve defined by the equations
and locate any critical points on its graph.
Integrals Involving Parametric Equations
Now that we have seen how to calculate the derivative of a plane curve, the next question is this: How do we find the area under a curve defined parametrically? Recall the cycloid defined by the equations Suppose we want to find the area of the shaded region in the following graph.
Figure 7.21Graph of a cycloid with the arch over highlighted.
To derive a formula for the area under the curve defined by the functions
we assume that is differentiable and start with an equal partition of the interval Suppose and consider the following graph.
Figure 7.22Approximating the area under a parametrically defined curve.
We use rectangles to approximate the area under the curve. The height of a typical rectangle in this parametrization is for some value in the ith subinterval, and the width can be calculated as Thus the area of the ith rectangle is given by
Then a Riemann sum for the area is
Multiplying and dividing each area by gives
Taking the limit as approaches infinity gives
This leads to the following theorem.
Theorem 7.2
Area under a Parametric Curve
Consider the non-self-intersecting plane curve defined by the parametric equations
and assume that is differentiable. The area under this curve is given by
7.3
Example 7.7
Finding the Area under a Parametric Curve
Find the area under the curve of the cycloid defined by the equations
Find the area under the curve of the hypocycloid defined by the equations
Arc Length of a Parametric Curve
In addition to finding the area under a parametric curve, we sometimes need to find the arc length of a parametric curve. In the case of a line segment, arc length is the same as the distance between the endpoints. If a particle travels from point A to point B along a curve, then the distance that particle travels is the arc length. To develop a formula for arc length, we start with an approximation by line segments as shown in the following graph.
Figure 7.23Approximation of a curve by line segments.
Given a plane curve defined by the functions we start by partitioning the interval into n equal subintervals: The width of each subinterval is given by We can calculate the length of each line segment:
Then add these up. We let s denote the exact arc length and denote the approximation by n line segments:
7.4
If we assume that and are differentiable functions of t, then the Mean Value Theorem (Introduction to the Applications of Derivatives) applies, so in each subinterval there exist and such that
This is a Riemann sum that approximates the arc length over a partition of the interval If we further assume that the derivatives are continuous and let the number of points in the partition increase without bound, the approximation approaches the exact arc length. This gives
When taking the limit, the values of and are both contained within the same ever-shrinking interval of width so they must converge to the same value.
We can summarize this method in the following theorem.
Theorem 7.3
Arc Length of a Parametric Curve
Consider the plane curve defined by the parametric equations
and assume that and are differentiable functions of t. Then the arc length of this curve is given by
7.5
At this point a side derivation leads to a previous formula for arc length. In particular, suppose the parameter can be eliminated, leading to a function Then and the Chain Rule gives Substituting this into Equation 7.5 gives
Here we have assumed that which is a reasonable assumption. The Chain Rule gives and letting and we obtain the formula
Find the arc length of the semicircle defined by the equations
Solution
The values to trace out the red curve in Figure 7.23. To determine its length, use Equation 7.5:
Note that the formula for the arc length of a semicircle is and the radius of this circle is 3. This is a great example of using calculus to derive a known formula of a geometric quantity.
Figure 7.24The arc length of the semicircle is equal to its radius times
Checkpoint 7.8
Find the arc length of the curve defined by the equations
We now return to the problem posed at the beginning of the section about a baseball leaving a pitcher’s hand. Ignoring the effect of air resistance (unless it is a curve ball!), the ball travels a parabolic path. Assuming the pitcher’s hand is at the origin and the ball travels left to right in the direction of the positive x-axis, the parametric equations for this curve can be written as
where t represents time. We first calculate the distance the ball travels as a function of time. This distance is represented by the arc length. We can modify the arc length formula slightly. First rewrite the functions and using v as an independent variable, so as to eliminate any confusion with the parameter t:
Then we write the arc length formula as follows:
The variable v acts as a dummy variable that disappears after integration, leaving the arc length as a function of time t. To integrate this expression we can use a formula from Appendix A,
We set and This gives so Therefore
and
This function represents the distance traveled by the ball as a function of time. To calculate the speed, take the derivative of this function with respect to t. While this may seem like a daunting task, it is possible to obtain the answer directly from the Fundamental Theorem of Calculus:
Therefore
One third of a second after the ball leaves the pitcher’s hand, the distance it travels is equal to
This value is just over three quarters of the way to home plate. The speed of the ball is
This speed translates to approximately 95 mph—a major-league fastball.
Surface Area Generated by a Parametric Curve
Recall the problem of finding the surface area of a volume of revolution. In Curve Length and Surface Area, we derived a formula for finding the surface area of a volume generated by a function from to revolved around the x-axis:
We now consider a volume of revolution generated by revolving a parametrically defined curve around the x-axis as shown in the following figure.
Figure 7.25A surface of revolution generated by a parametrically defined curve.
The analogous formula for a parametrically defined curve is
7.6
provided that is not negative on
Example 7.9
Finding Surface Area
Find the surface area of a sphere of radius r centered at the origin.
Solution
We start with the curve defined by the equations
This generates an upper semicircle of radius r centered at the origin as shown in the following graph.
Figure 7.26A semicircle generated by parametric equations.
When this curve is revolved around the x-axis, it generates a sphere of radius r. To calculate the surface area of the sphere, we use Equation 7.6:
This is, in fact, the formula for the surface area of a sphere.
Checkpoint 7.9
Find the surface area generated when the plane curve defined by the equations
is revolved around the x-axis.
Section 7.2 Exercises
For the following exercises, each set of parametric equations represents a line. Without eliminating the parameter, find the slope of each line.
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