Calculus Volume 3

# Key Concepts

Calculus Volume 3Key Concepts

### 5.1Double Integrals over Rectangular Regions

• We can use a double Riemann sum to approximate the volume of a solid bounded above by a function of two variables over a rectangular region. By taking the limit, this becomes a double integral representing the volume of the solid.
• Properties of double integral are useful to simplify computation and find bounds on their values.
• We can use Fubini’s theorem to write and evaluate a double integral as an iterated integral.
• Double integrals are used to calculate the area of a region, the volume under a surface, and the average value of a function of two variables over a rectangular region.

### 5.2Double Integrals over General Regions

• A general bounded region $DD$ on the plane is a region that can be enclosed inside a rectangular region. We can use this idea to define a double integral over a general bounded region.
• To evaluate an iterated integral of a function over a general nonrectangular region, we sketch the region and express it as a Type I or as a Type II region or as a union of several Type I or Type II regions that overlap only on their boundaries.
• We can use double integrals to find volumes, areas, and average values of a function over general regions, similarly to calculations over rectangular regions.
• We can use Fubini’s theorem for improper integrals to evaluate some types of improper integrals.

### 5.3Double Integrals in Polar Coordinates

• To apply a double integral to a situation with circular symmetry, it is often convenient to use a double integral in polar coordinates. We can apply these double integrals over a polar rectangular region or a general polar region, using an iterated integral similar to those used with rectangular double integrals.
• The area $dAdA$ in polar coordinates becomes $rdrdθ.rdrdθ.$
• Use $x=rcosθ,x=rcosθ,$ $y=rsinθ,y=rsinθ,$ and $dA=rdrdθdA=rdrdθ$ to convert an integral in rectangular coordinates to an integral in polar coordinates.
• Use $r2=x2+y2r2=x2+y2$ and $θ=tan−1(yx)θ=tan−1(yx)$ to convert an integral in polar coordinates to an integral in rectangular coordinates, if needed.
• To find the volume in polar coordinates bounded above by a surface $z=f(r,θ)z=f(r,θ)$ over a region on the $xyxy$-plane, use a double integral in polar coordinates.

### 5.4Triple Integrals

• To compute a triple integral we use Fubini’s theorem, which states that if $f(x,y,z)f(x,y,z)$ is continuous on a rectangular box $B=[a,b]×[c,d]×[e,f],B=[a,b]×[c,d]×[e,f],$ then
$∭Bf(x,y,z)dV=∫ef∫cd∫abf(x,y,z)dxdydz∭Bf(x,y,z)dV=∫ef∫cd∫abf(x,y,z)dxdydz$

and is also equal to any of the other five possible orderings for the iterated triple integral.
• To compute the volume of a general solid bounded region $EE$ we use the triple integral
$V(E)=∭E1dV.V(E)=∭E1dV.$
• Interchanging the order of the iterated integrals does not change the answer. As a matter of fact, interchanging the order of integration can help simplify the computation.
• To compute the average value of a function over a general three-dimensional region, we use
$fave=1V(E)∭Ef(x,y,z)dV.fave=1V(E)∭Ef(x,y,z)dV.$

### 5.5Triple Integrals in Cylindrical and Spherical Coordinates

• To evaluate a triple integral in cylindrical coordinates, use the iterated integral
$∫θ=αθ=β∫r=g1(θ)r=g2(θ)∫z=u1(r,θ)z=u2(r,θ)f(r,θ,z)rdzdrdθ.∫θ=αθ=β∫r=g1(θ)r=g2(θ)∫z=u1(r,θ)z=u2(r,θ)f(r,θ,z)rdzdrdθ.$
• To evaluate a triple integral in spherical coordinates, use the iterated integral
$∫θ=αθ=β∫ρ=g1(θ)ρ=g2(θ)∫φ=u1(r,θ)φ=u2(r,θ)f(ρ,θ,φ)ρ2sinφdφdρdθ.∫θ=αθ=β∫ρ=g1(θ)ρ=g2(θ)∫φ=u1(r,θ)φ=u2(r,θ)f(ρ,θ,φ)ρ2sinφdφdρdθ.$

### 5.6Calculating Centers of Mass and Moments of Inertia

Finding the mass, center of mass, moments, and moments of inertia in double integrals:

• For a lamina $RR$ with a density function $ρ(x,y)ρ(x,y)$ at any point $(x,y)(x,y)$ in the plane, the mass is $m=∬Rρ(x,y)dA.m=∬Rρ(x,y)dA.$
• The moments about the $x-axisx-axis$ and $y-axisy-axis$ are
$Mx=∬Ryρ(x,y)dAandMy=∬Rxρ(x,y)dA.Mx=∬Ryρ(x,y)dAandMy=∬Rxρ(x,y)dA.$
• The center of mass is given by $x−=Mym,y−=Mxm.x−=Mym,y−=Mxm.$
• The center of mass becomes the centroid of the plane when the density is constant.
• The moments of inertia about the $x−axis,x−axis,$ $y−axis,y−axis,$ and the origin are
$Ix=∬Ry2ρ(x,y)dA,Iy=∬Rx2ρ(x,y)dA,andI0=Ix+Iy=∬R(x2+y2)ρ(x,y)dA.Ix=∬Ry2ρ(x,y)dA,Iy=∬Rx2ρ(x,y)dA,andI0=Ix+Iy=∬R(x2+y2)ρ(x,y)dA.$

Finding the mass, center of mass, moments, and moments of inertia in triple integrals:

• For a solid object $QQ$ with a density function $ρ(x,y,z)ρ(x,y,z)$ at any point $(x,y,z)(x,y,z)$ in space, the mass is $m=∭Qρ(x,y,z)dV.m=∭Qρ(x,y,z)dV.$
• The moments about the $xy-plane,xy-plane,$ the $xz-plane,xz-plane,$ and the $yz-planeyz-plane$ are
$Mxy=∭Qzρ(x,y,z)dV,Mxz=∭Qyρ(x,y,z)dV,Myz=∭Qxρ(x,y,z)dV.Mxy=∭Qzρ(x,y,z)dV,Mxz=∭Qyρ(x,y,z)dV,Myz=∭Qxρ(x,y,z)dV.$
• The center of mass is given by $x−=Myzm,y−=Mxzm,z−=Mxym.x−=Myzm,y−=Mxzm,z−=Mxym.$
• The center of mass becomes the centroid of the solid when the density is constant.
• The moments of inertia about the $yz-plane,yz-plane,$ the $xz-plane,xz-plane,$ and the $xy-planexy-plane$ are
$Ix=∭Q(y2+z2)ρ(x,y,z)dV,Iy=∭Q(x2+z2)ρ(x,y,z)dV,Iz=∭Q(x2+y2)ρ(x,y,z)dV.Ix=∭Q(y2+z2)ρ(x,y,z)dV,Iy=∭Q(x2+z2)ρ(x,y,z)dV,Iz=∭Q(x2+y2)ρ(x,y,z)dV.$

### 5.7Change of Variables in Multiple Integrals

• A transformation $TT$ is a function that transforms a region $GG$ in one plane (space) into a region $RR$ in another plane (space) by a change of variables.
• A transformation $T:G→RT:G→R$ defined as $T(u,v)=(x,y)T(u,v)=(x,y)$ $(orT(u,v,w)=(x,y,z))(orT(u,v,w)=(x,y,z))$ is said to be a one-to-one transformation if no two points map to the same image point.
• If $ff$ is continuous on $R,R,$ then $∬Rf(x,y)dA=∬Sf(g(u,v),h(u,v))|∂(x,y)∂(u,v)|dudv.∬Rf(x,y)dA=∬Sf(g(u,v),h(u,v))|∂(x,y)∂(u,v)|dudv.$
• If $FF$ is continuous on $R,R,$ then
$∭RF(x,y,z)dV=∭GF(g(u,v,w),h(u,v,w),k(u,v,w))|∂(x,y,z)∂(u,v,w)|dudvdw=∭GH(u,v,w)|J(u,v,w)|dudvdw.∭RF(x,y,z)dV=∭GF(g(u,v,w),h(u,v,w),k(u,v,w))|∂(x,y,z)∂(u,v,w)|dudvdw=∭GH(u,v,w)|J(u,v,w)|dudvdw.$