Chapter 15

Use a CAS double-integral evaluator to estimate the values of the integrals in Exercises \(85-88 .\) \begin{equation}\int_{1}^{3} \int_{1}^{x} \frac{1}{x y} d y d x\end{equation}

Sphere and paraboloid Find the volume of the region bounded above by the sphere \(x^{2}+y^{2}+z^{2}=2\) and below by the paraboloid \(z=x^{2}+y^{2}\)

Use a CAS double-integral evaluator to estimate the values of the integrals. \begin{equation}\int_{0}^{1} \int_{0}^{1} e^{-\left(x^{2}+y^{2}\right)} d y d x\end{equation}

Find the average value of the function \(f(r, \theta, z)=r\) over the solid ball bounded by the sphere \(r^{2}+z^{2}=1 .\) (This is the sphere \(x^{2}+y^{2}+z^{2}=1 . )\)

Use a CAS double-integral evaluator to estimate the values of the integrals. \begin{equation}\int_{0}^{1} \int_{0}^{1} \tan ^{-1} x y d y d x\end{equation}

Use a CAS double-integral evaluator to estimate the values of the integrals. \begin{equation}\int_{-1}^{1} \int_{0}^{\sqrt{1-x^{2}}} 3 \sqrt{1-x^{2}-y^{2}} d y d x\end{equation}

Find the average value of the function \(f(\rho, \phi, \theta)=\rho \cos \phi\) over the solid upper ball \(\rho \leq 1,0 \leq \phi \leq \pi / 2\)

Use a CAS double-integral evaluator to find the integrals in Exercises \(89-94\) . Then reverse the order of integration and evaluate, again with a CAS. \begin{equation}\int_{0}^{1} \int_{2 y}^{4} e^{x^{2}} d x d y\end{equation}

Center of mass A solid of constant density is bounded below by the plane \(z=0,\) above by the cone \(z=r, r \geq 0,\) and on the sides by the cylinder \(r=1 .\) Find the center of mass.

Use a CAS double-integral evaluator to find the integrals . Then reverse the order of integration and evaluate, again with a CAS. \begin{equation}\int_{0}^{3} \int_{x^{2}}^{9} x \cos \left(y^{2}\right) d y d x\end{equation}

Centroid Find the centroid of the region in the first octant that is bounded above by the cone \(z=\sqrt{x^{2}+y^{2}}\) , below by the plane \(z=0,\) and on the sides by the cylinder \(x^{2}+y^{2}=4\) and the planes \(x=0\) and \(y=0\)

Use a CAS double-integral evaluator to find the integrals . Then reverse the order of integration and evaluate, again with a CAS. \begin{equation}\int_{0}^{2} \int_{y^{3}}^{4 \sqrt{2 y}}\left(x^{2} y-x y^{2}\right) d x d y\end{equation}

Use a CAS double-integral evaluator to find the integrals . Then reverse the order of integration and evaluate, again with a CAS. \begin{equation}\int_{0}^{2} \int_{0}^{4-y^{2}} e^{x y} d x d y\end{equation}

Centroid Find the centroid of the solid bounded above by the sphere \(\rho=a\) and below by the cone \(\phi=\pi / 4 .\)

Use a CAS double-integral evaluator to find the integrals . Then reverse the order of integration and evaluate, again with a CAS. \begin{equation}\int_{1}^{2} \int_{0}^{x^{2}} \frac{1}{x+y} d y d x\end{equation}

Centroid Find the centroid of the region that is bounded above by the surface \(z=\sqrt{r},\) on the sides by the cylinder \(r=4,\) and below by the \(x y\) -plane.

Use a CAS double-integral evaluator to find the integrals . Then reverse the order of integration and evaluate, again with a CAS. \begin{equation}\int_{1}^{2} \int_{y^{3}}^{8} \frac{1}{\sqrt{x^{2}+y^{2}}} d x d y\end{equation}

Centroid Find the centroid of the region cut from the solid ball \(r^{2}+z^{2} \leq 1\) by the half-planes \(\theta=-\pi / 3, r \geq 0,\) and \(\theta=\pi / 3, r \geq 0\)

Moment of inertia of solid cone Find the moment of inertia of a right circular cone of base radius 1 and height 1 about an axis through the vertex parallel to the base. (Take \(\delta=1 . )\)

Moment of inertia of solid sphere Find the moment of inertia of a solid sphere of radius \(a\) about a diameter. (Take \(\delta=1 . )\)

Moment of inertia of solid cone Find the moment of inertia of a right circular cone of base radius \(a\) and height \(h\) about its axis. (Hint: Place the cone with its vertex at the origin and its axis along the \(z\) -axis.)

Variable density A solid is bounded on the top by the paraboloid \(z=r^{2},\) on the bottom by the plane \(z=0,\) and on the sides by the cylinder \(r=1 .\) Find the center of mass and the moment of inertia about the \(z\) -axis if the density is $$\begin{array}{ll}{\text { a. }} & {\delta(r, \theta, z)=z} \\ {\text { b. }} & {\delta(r, \theta, z)=r}\end{array}$$

Variable density \(A\) solid is bounded below by the cone \(z=\sqrt{x^{2}+y^{2}}\) and above by the plane \(z=1 .\) Find the center of mass and the moment of inertia about the \(z\) -axis if the density is $$\begin{array}{ll}{\text { a. }} & {\delta(r, \theta, z)=z} \\ {\text { b. }} & {\delta(r, \theta, z)=z^{2}}\end{array}$$

Variable density A solid ball is bounded by the sphere \(\rho=a\) . Find the moment of inertia about the \(z\) -axis if the density is $$\begin{aligned} \text { a. } & \delta(\rho, \phi, \theta)=\rho^{2} \\ \text { b. } & \delta(\rho, \phi, \theta)=r=\rho \sin \phi \end{aligned}$$

Centroid of solid semiellipsoid Show that the centroid of the solid semiellipsoid of revolution \(\left(r^{2} / a^{2}\right)+\left(z^{2} / h^{2}\right) \leq 1, z \geq 0\) lies on the \(z\) -axis three-eighths of the way from the base to the top. The special case \(h=a\) gives a solid hemisphere. Thus, the centroid of a solid hemisphere lies on the axis of symmetry three-eighths of the way from the base to the top.

Centroid of solid cone Show that the centroid of a solid right circular cone is one-fourth of the way from the base to the vertex. (In general, the centroid of a solid cone or pyramid is one-fourth of the way from the centroid of the base to the vertex.)

Density of center of a planet \(A\) planet is in the shape of a sphere of radius \(R\) and total mass \(M\) with spherically symmetric density distribution that increases linearly as one approaches its center. What is the density at the center of this planet if the density at its edge (surface) is taken to be zero?

Mass of planet's atmosphere \(A\) spherical planet of radius \(R\) has an atmosphere whose density is \(\mu=\mu_{0} e^{-c h},\) where \(h\) is the altitude above the surface of the planet, \(\mu_{0}\) is the density at sea level, and \(c\) is a positive constant. Find the mass of the planet's atmosphere.

Vertical planes in cylindrical coordinates a. Show that planes perpendicular to the \(x\) -axis have equations of the form \(r=a \sec \theta\) in cylindrical coordinates. b. Show that planes perpendicular to the \(y\) -axis have equations of the form \(r=b \csc \theta .\)

Symmetry What symmetry will you find in a surface that has an equation of the form \(r=f(z)\) in cylindrical coordinates? Give reasons for your answer.

Symmetry What symmetry will you find in a surface that has an equation of the form \(\rho=f(\phi)\) in spherical coordinates? Give reasons for your answer.