Chapter 13

In Problems 15-20, an iterated integral is given either in rectangular or polar coordinates. The double integral gives the mass of some lamina \(R\). Sketch the lamina \(R\) and determine the density \(\delta\). Then find the mass and center of mass. \(\int_{0}^{\pi / 2} \int_{0}^{\theta} k r d r d \theta\)

In Problems 15-22, use spherical coordinates to find the indicated quantity. Volume of the smaller wedge cut from the unit sphere by two planes that meet at a diameter at an angle of \(30^{\circ}\)

In Problems 21-32, sketch the indicated solid. Then find its volume by an iterated integration. Tetrahedron bounded by the coordinate planes and the plane \(z=6-2 x-3 y\)

Find the Jacobian for the transformation from rectangular coordinates to spherical coordinates.

Volume of the solid in the first octant bounded by \(y=2 x^{2}\) and \(y+4 z=8\)

In Problems 19-26, evaluate by using polar coordinates. Sketch the region of integration first. \(\iint_{S} y d A\), where \(S\) is the first quadrant polar rectangle inside \(x^{2}+y^{2}=4\) and outside \(x^{2}+y^{2}=1\)

In Problems 21-32, sketch the indicated solid. Then find its volume by an iterated integration. Tetrahedron bounded by the coordinate planes and the plane \(3 x+4 y+z-12=0\)

Find the volume of the ellipsoid \(x^{2} / a^{2}+y^{2} / b^{2}+\) \(z^{2} / c^{2}=1\) by making the change of variables \(x=u a, y=v b\), and \(z=c w\). Also, find the moment of inertia of this solid about the \(z\)-axis assuming that it has constant density \(k\).

Calculate \(\iint_{R}(1+x) d A\), where \(R=\\{(x, y): 0 \leq x \leq 2\), \(0 \leq y \leq 1\\} .\) See the hint in Problem \(21 .\)

Find the volume of the solid bounded above by the plane \(z=y\) and below by the paraboloid \(z=x^{2}+y^{2}\). Hint: In cylindrical coordinates the plane has equation \(z=r \sin \theta\) and the paraboloid has equation \(z=r^{2}\). Solve simultaneously to get the projection in the \(x y\)-plane.

In Problems 21-32, sketch the indicated solid. Then find its volume by an iterated integration. Wedge bounded by the coordinate planes and the planes \(x=5\) and \(y+2 z-4=0\)

Suppose \(X\) and \(Y\) are continuous random variables with joint PDF \(f(x, y)\) and suppose \(U\) and \(V\) are random variables that are functions of \(X\) and \(Y\) such that the transformation $$ X=x(U, V) \quad \text { and } \quad Y=y(U, V) $$ is one-to-one. Show that the joint PDF of \(U\) and \(V\) is $$ g(u, v)=f(x(u, v), y(u, v))|J(u, v)| $$ Hint: Let \(R\) be a region in the \(x y\)-plane and let \(S\) be its preimage. Show that \(P((X, Y) \in R)=P((U, V) \in S)\) and get a double integral for each of these.

Use the comparison property of double integrals to show that if \(f(x, y) \geq 0\) on \(R\) then \(\iint_{R} f(x, y) d A \geq 0\).

Find the moment of inertia and radius of gyration of a homogeneous ( \(\delta\) a constant) circular lamina of radius \(a\) with respect to a diameter.

Volume of the solid bounded by the cylinders \(x^{2}=y\) and \(z^{2}=y\) and the plane \(y=1\)

Find the volume of the solid inside both of the spheres \(\rho=2 \sqrt{2} \cos \phi\) and \(\rho=2\).

In Problems 21-32, sketch the indicated solid. Then find its volume by an iterated integration. Solid in the first octant bounded by the coordinate planes and the planes \(2 x+y-4=0\) and \(8 x+y-4 z=0\)

Suppose that the random variables \(X\) and \(Y\) have joint PDF $$ f(x, y)= \begin{cases}\frac{1}{4}, & \text { if } 0 \leq x \leq 2,0 \leq y \leq 2 \\ 0, & \text { otherwise }\end{cases} $$ that is, \(X\) and \(Y\) are uniformly distributed over the square \(0 \leq x \leq 2,0 \leq y \leq 2\). Find (a) the joint PDF of \(U=X+Y\) and \(V=X-Y\), and (b) the marginal PDF of \(U\).

Suppose that \(m \leq f(x, y) \leq M\) on \(R\). Show that $$ m A(R) \leq \iint_{R} f(x, y) d A \leq M A(R) $$

Show that the moment of inertia of a homogeneous rectangular lamina with sides of length \(a\) and \(b\) about a perpendicular axis through its center of mass is $$ I=\frac{1}{12}\left(a^{3} b+a b^{3}\right) $$ Here \(k\) is the constant density.

Volume of the solid bounded by the cylinder \(y=x^{2}+2\) and the planes \(y=4, z=0\), and \(3 y-4 z=0\)

In Problems 19-26, evaluate by using polar coordinates. Sketch the region of integration first. $$ \int_{0}^{1} \int_{x}^{1} x^{2} d y d x $$

For a solid sphere of radius \(a\), find each average distance. (a) From its center (b) From a diameter (c) From a point on its boundary (consider \(\rho=2 a \cos \phi\) )

In Problems 21-32, sketch the indicated solid. Then find its volume by an iterated integration. Solid in the first octant bounded by the surface \(9 x^{2}+4 y^{2}=36\) and the plane \(9 x+4 y-6 z=0\)

Suppose \(X\) and \(Y\) have joint PDF $$ f(x, y)= \begin{cases}e^{-x-y}, & \text { if } x \geq 0, y \geq 0 \\ 0, & \text { otherwise }\end{cases} $$ Find (a) the joint PDF of \(U=X+Y\) and \(V=X\) (b) the marginal PDF of \(U\).

Sketch the solid whose volume is the indicated iterated integral. \(\int_{0}^{1} \int_{0}^{2} \frac{x}{2} d x d y\)

Center of mass of the tetrahedron bounded by the planes \(x+y+z=1, x=0, y=0\), and \(z=0\) if the density is proportional to the sum of the coordinates of the point

In Problems 19-26, evaluate by using polar coordinates. Sketch the region of integration first. $$ \int_{1}^{2} \int_{0}^{\sqrt{2 x-x^{2}}}\left(x^{2}+y^{2}\right)^{-1 / 2} d y d x $$

In Problems 21-32, sketch the indicated solid. Then find its volume by an iterated integration. Solid in the first octant bounded by the surface \(z=9-x^{2}-y^{2}\) and the coordinate planes

Sketch the solid whose volume is the indicated iterated integral. \(\int_{0}^{1} \int_{0}^{1}(2-x-y) d y d x\)

Let \(\gamma=\gamma(x, y, f(x, y))\) be the acute angle between the \(z\)-axis and a normal vector to the surface \(z=f(x, y)\) at the point \((x, y, f(x, y))\) on the surface. Show that sec \(\gamma=\sqrt{f_{x}^{2}+f_{y}^{2}+1}\). (Note that this gives another formula for surface area: \(A(G)=\) \(\left.\iint_{S} \sec \gamma d A .\right)\)

Center of mass of the solid bounded by the cylinder \(x^{2}+y^{2}=9\) and the planes \(z=0\) and \(z=4\) if the density is proportional to the square of the distance from the origin

Find the volume of the solid in the first octant under the paraboloid \(z=x^{2}+y^{2}\) and inside the cylinder \(x^{2}+y^{2}=9\) by using polar coordinates.

In Problems 21-32, sketch the indicated solid. Then find its volume by an iterated integration. Solid in the first octant bounded by the cylinder \(y=x^{2}\) and the planes \(x=0, z=0\), and \(y+z=1\)

The paraboloid \(z=x^{2}+y^{2}\) over the region (a) in the first quadrant and inside the circle \(x^{2}+y^{2}=9\) (b) inside the triangle with vertices \((0,0),(3,0),(0,3)\)

Center of mass of that part of the solid sphere \(\\{(x, y, z)\) : \(\left.x^{2}+y^{2}+z^{2} \leq a^{2}\right\\}\) that lies in the first octant, assuming that it has constant density

Using polar coordinates, find the volume of the solid bounded above by \(2 x^{2}+2 y^{2}+z^{2}=18\), below by \(z=0\), and laterally by \(x^{2}+y^{2}=4\).

In Problems 21-32, sketch the indicated solid. Then find its volume by an iterated integration. Solid bounded by the parabolic cylinder \(x^{2}=4 y\) and the planes \(z=0\) and \(5 y+9 z-45=0\)

Sketch the solid whose volume is the indicated iterated integral. \(\int_{0}^{2} \int_{0}^{2}\left(4-y^{2}\right) d y d x\)

The hyperbolic paraboloid \(z=y^{2}-x^{2}\) over the region (a) in the first quadrant and inside the circle \(x^{2}+y^{2}=9\) (b) inside the triangle with vertices \((0,0),(3,0),(0,3)\)

Parallel Axis Theorem Consider a lamina \(S\) of mass \(m\) together with parallel lines \(L\) and \(L^{\prime}\) in the plane of \(S\), the line \(L\) passing through the center of mass of \(S\). Show that if \(I\) and \(I^{\prime}\) are the moments of inertia of \(S\) about \(L\) and \(L^{\prime}\), respectively, then \(I^{\prime}=I+d^{2} m\), where \(d\) is the distance between \(L\) and \(L^{\prime}\). Hint: Assume that \(S\) lies in the \(x y\)-plane, \(L\) is the \(y\)-axis, and \(L^{\prime}\) is the line \(x=-d\).

Moment of inertia \(I_{x}\) about the \(x\)-axis of the solid bounded by the cylinder \(y^{2}+z^{2}=4\) and the planes \(x-y=0, x=0\), and \(z=0\) if the density \(\delta(x, y, z)=z\). Hint: You will need to develop your own formula; slice, approximate, integrate.

Colorado is a rectangular state (if we ignore the curvature of the earth). Let \(f(x, y)\) be the number of inches of rainfall during 2005 at the point \((x, y)\) in that state. What does \(\iint_{\text {Colorado }} f(x, y) d A\) represent? What does this number divided by the area of Colorado represent?

Find the volume of the given solid. First, sketch the solid; then estimate its volume; finally, determine its exact volume. Solid under the plane \(z=x+y+1\) over \(R=\\{(x, y)\) : \(0 \leq x \leq 1,1 \leq y \leq 3\\}\)

Six surfaces are given below. Without performing any integration, rank the surfaces in order of their surface area from smallest to largest. Hint: There may be some "ties." (a) The paraboloid \(z=x^{2}+y^{2}\) over the region in the first quadrant and inside the circle \(x^{2}+y^{2}=1\) (b) The hyperbolic paraboloid \(z=x^{2}-y^{2}\) over the region in the first quadrant and inside the circle \(x^{2}+y^{2}=1\) (c) The paraboloid \(z=x^{2}+y^{2}\) over the region inside the rectangle with vertices \((0,0),(1,0),(1,1)\), and \((0,1)\) (d) The hyperbolic paraboloid \(z=x^{2}-y^{2}\) over the region inside the rectangle with vertices \((0,0),(1,0),(1,1)\), and \((0,1)\) (e) The paraboloid \(z=x^{2}+y^{2}\) over the region inside the triangle with vertices \((0,0),(1,0)\), and \((0,1)\) (f) The hyperbolic paraboloid \(z=x^{2}-y^{2}\) over the region inside the triangle with vertices \((0,0),(1,0)\), and \((0,1)\)

In Problems 29-32, write the given iterated integral as an iterated integral with the indicated order of integration. $$ \int_{0}^{1} \int_{0}^{\sqrt{1-y^{2}}} \int_{0}^{\sqrt{1-y^{2}-z^{2}}} f(x, y, z) d x d z d y ; d z d y d x $$

In Problems 21-32, sketch the indicated solid. Then find its volume by an iterated integration. Solid in the first octant bounded by the surface \(z=e^{x-y}\), the plane \(x+y=1\), and the coordinate planes

Find the volume of the given solid. First, sketch the solid; then estimate its volume; finally, determine its exact volume. Solid under the plane \(z=2 x+3 y\) and over \(R=\\{(x, y)\) : \(1 \leq x \leq 2,0 \leq y \leq 4\\}\)

In Problems 29-32, write the given iterated integral as an iterated integral with the indicated order of integration. $$ \int_{0}^{2} \int_{0}^{4-2 y} \int_{0}^{4-2 y-z} f(x, y, z) d x d z d y ; d z d y d x $$

The centers of two spheres of radius \(a\) are \(2 b\) units apart with \(b \leq a\). Find the volume of their intersection in terms of \(d=a-b\).