Chapter 13

Use triple iterated integrals to find the indicated quantities. Volume of the solid in the first octant bounded by \(y=2 x^{2}\) and \(y+4 z=8\)

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 \(z=6-2 x-3 y\)

Calculate \(\iint_{R}(6-y) d A\), where \(R=\\{(x, y): 0 \leq x \leq 1\), \(0 \leq y \leq 1\\} .\) Hint: This integral represents the volume of a certain solid. Sketch this solid and calculate its volume from elementary principles.

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 \(k\).

Use triple iterated integrals to find the indicated quantities. Volume of the solid in the first octant bounded by the elliptic cylinder \(y^{2}+64 z^{2}=4\) and the plane \(y=x\)

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 \(3 x+4 y+z-12=0\)

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

Suppose \(X\) and \(Y\) are continuous random variables with joint \(\operatorname{PDF} f(x, y)\) 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 \(\mathrm{PDF}\) 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 triple iterated integrals to find the indicated quantities. Volume of the solid bounded by the cylinders \(x^{2}=y\) and \(z^{2}=y\) and the plane \(y=1\)

Evaluate by using polar coordinates. Sketch the region of integration first. $$ \int_{0}^{1} \int_{0}^{\sqrt{1-x^{2}}}\left(4-x^{2}-y^{2}\right)^{-1 / 2} d y d x $$

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

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

Use spherical coordinates to find the indicated quantity. Find the volume of the solid inside both of the spheres \(\rho=2 \sqrt{2} \cos \phi\) and \(\rho=2\).

Suppose that the random variables \(X\) and \(Y\) have joint PDF $$ f(x, y)=\left\\{\begin{array}{ll} \frac{1}{4}, & \text { if } 0 \leq x \leq 2,0 \leq y \leq 2 \\ 0, & \text { otherwise } \end{array}\right. $$ 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 \(U\). 25\. Suppose \(X\) and \(Y\) have joint \(\mathrm{PDF}\) $$ f(x, y)=\left\\{\begin{array}{ll} e^{-x-y}, & \text { if } x \geq 0, y \geq 0 \\ 0, & \text { otherwise } \end{array}\right. $$ Find (a) the joint PDF of \(U=X+Y\) and \(V=X\) (b) the marginal PDF of \(U\).

Evaluate by using polar coordinates. Sketch the region of integration first. $$ \int_{0}^{1} \int_{0}^{\sqrt{1-y^{2}}} \sin \left(x^{2}+y^{2}\right) d x d y $$

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

Use spherical coordinates to find the indicated quantity. 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 \(\rho=2 a \cos \phi\) )

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

Show that the surface area of a nonvertical plane over a region \(S\) in the \(x y\) -plane is \(A(S) \sec \gamma\) where \(\gamma\) is the acute angle between a normal vector to the plane and the positive \(z\) -axis.

Use triple iterated integrals to find the indicated quantities. 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

Evaluate by using polar coordinates. Sketch the region of integration first. $$ \int_{0}^{1} \int_{x}^{1} x^{2} d y d x $$

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 \(9 x+4 y-6 z=0\)

Use spherical coordinates to find the indicated quantity. For any homogeneous solid \(S\), show that the average value of the linear function \(f(x, y, z)=a x+b y+c z+d\) on \(S\) is \(f(\bar{x}, \bar{y}, \bar{z})\), where \((\bar{x}, \bar{y}, \bar{z})\) is the center of mass.

Use triple iterated integrals to find the indicated quantities. 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

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. $$ \int_{0}^{1} \int_{0}^{1}(2-x-y) d y d x $$

Use spherical coordinates to find the indicated quantity. A homogeneous solid sphere of radius \(a\) is centered at the origin. For the section \(S\) bounded by the half-planes \(\theta=-\alpha\) and \(\theta=\alpha\) (like a section of an orange), find each value. (a) \(x\) -coordinate of the center of mass (b) Average distance from the \(z\) -axis

Find the surface area of the given surface. If an integral cannot be evaluated using the Second Fundamental Theorem of Calculus, then use the Parabolic Rule with \(n=10 .\) 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 \((0,0),(3,0),(0,3)\)

Use triple iterated integrals to find the indicated quantities. 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

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.

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 \(y+z=1\)

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

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 \(x=-d\).

Use triple iterated integrals to find the indicated quantities. 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.

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 \(x^{2}+y^{2}=4\).

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. $$ \int_{0}^{2} \int_{0}^{2}\left(4-y^{2}\right) d y d x $$

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

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 \((0,1)\)

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 \(f(x, y) d A\) represent? What does this number divided by Colorado 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 \(R=\\{(x, y)\) : \(0 \leq x \leq 1,1 \leq y \leq 3\\}\)

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

Let \(V=\iint_{S} \sin \sqrt{x^{2}+y^{2}} d A\) and \(W=\) \(\iint_{S}\left|\sin \sqrt{x^{2}+y^{2}}\right| d A\), where \(S\) is the region inside the circle \(x^{2}+y^{2}=4 \pi^{2} .\) (a) Without calculation, determine the sign of \(V\). (b) Evaluate \(V\). (c) Evaluate \(W\).

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 \(R=\\{(x, y)\) : \(1 \leq x \leq 2,0 \leq y \leq 4\\}\)

Write the given iterated integral as an iterated integral with the indicated order of integration. \(\int_{0}^{2} \int_{0}^{9-x^{2}} \int_{0}^{2-x} f(x, y, z) d z d y d x ; d y d x d z\)

Sketch the indicated solid. Then find its volume by an iterated integration. Solid in the first octant bounded by the circular cylinders \(x^{2}+z^{2}=16\) and \(y^{2}+z^{2}=16\) and the coordinate planes