Chapter 14

Find the volume of the region bounded below by the paraboloid \(z=x^{2}+y^{2},\) laterally by the cylinder \(x^{2}+y^{2}=1,\) and above by the paraboloid \(z=\) \(x^{2}+y^{2}+1\).

Sketch the region of integration, reverse the order of integration, and evaluate the integral. $$\int_{0}^{8} \int_{\sqrt{x}}^{2} \frac{d y d x}{y^{4}+1}$$

Find the volume of the region bounded below by the paraboloid \(z=x^{2}+y^{2},\) laterally by the cylinder \(x^{2}+y^{2}=1,\) and above by the paraboloid \(z=\) \(x^{2}+y^{2}+1\).

Sketch the region of integration, reverse the order of integration, and evaluate the integral. Square region \(\iint_{R}\left(y-2 x^{2}\right) d A\) where \(R\) is the region bounded by the square \(|x|+|y|=1\)

Find the volume of the region that lies inside the sphere \(x^{2}+y^{2}+z^{2}=2\) and outside the cylinder \(x^{2}+y^{2}=1\).

Sketch the region of integration, reverse the order of integration, and evaluate the integral. Triangular region \(\iint_{R} x y d A\) where \(R\) is the region bounded by the lines \(y=x, y=2 x,\) and \(x+y=2\)

Find the volume of the region enclosed by the cylinder \(x^{2}+y^{2}=4\) and the planes \(z=0\) and \(y+z=4\).

Find the volume of the region bounded above by the paraboloid \(z=x^{2}+y^{2}\) and below by the triangle enclosed by the lines \(y=x, x=0,\) and \(x+y=2\) in the \(x y\) -plane.

Find the volume of the region enclosed by the cylinder \(x^{2}+y^{2}=4\) and the planes \(z=0\) and \(x+y+z=4\).

Sketch the region of integration, reverse the order of integration, and evaluate the integral. Find the volume of the solid that is bounded above by the cylinder \(z=x^{2}\) and below by the region enclosed by the parabola \(y=2-x^{2}\) and the line \(y=x\) in the \(x y\) -plane.

Find the volume of the region bounded above by the paraboloid \(z=5-x^{2}-y^{2}\) and below by the paraboloid \(z=4 x^{2}+4 y^{2}\).

Sketch the region of integration, reverse the order of integration, and evaluate the integral. Find the volume of the solid whose base is the region in the \(x y-\) plane that is bounded by the parabola \(y=4-x^{2}\) and the line \(y=3 x,\) while the top of the solid is bounded by the plane \(z=x+4\)

Find the volume of the region bounded above by the paraboloid \(z=9-x^{2}-y^{2},\) below by the \(x y\) -plane, and lying outside the cylinder \(x^{2}+y^{2}=1\).

Sketch the region of integration, reverse the order of integration, and evaluate the integral. Find the volume of the solid in the first octant bounded by the coordinate planes, the cylinder \(x^{2}+y^{2}=4,\) and the plane \(z+y=3\)

Find the volume of the region cut from the solid cylinder \(x^{2}+y^{2} \leq 1\) by the sphere \(x^{2}+y^{2}+z^{2}=4\).

Sketch the region of integration, reverse the order of integration, and evaluate the integral. Find the volume of the solid in the first octant bounded by the coordinate planes, the plane \(x=3,\) and the parabolic cylinder \(z=4-y^{2}\)

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

Find the average value of the function \(f(r, \theta, z)=r\) over the region bounded by the cylinder \(r=1\) between the planes \(z=-1\) and \(z=1\).

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 \(\left.x^{2}+y^{2}+z^{2}=1 .\right)\)

Sketch the region of integration, reverse the order of integration, and evaluate the integral. Find the volume of the solid cut from the square column \(|x|+|y| \leq 1\) by the planes \(z=0\) and \(3 x+z=3\)

Find the average value of the function \(f(\rho, \phi, \theta)=\rho\) over the solid ball \(\rho \leq 1\).

Sketch the region of integration, reverse the order of integration, and evaluate the integral. Find the volume of the solid that is bounded on the front and back by the planes \(x=2\) and \(x=1,\) on the sides by the cylinders \(y=\pm 1 / x,\) and above and below by the planes \(z=x+1\) and \(z=0\)

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

Sketch the region of integration, reverse the order of integration, and evaluate the integral. Find the volume of the solid bounded on the front and back by the planes \(x=\pm \pi / 3,\) on the sides by the cylinders \(y=\pm \sec x\) above by the cylinder \(z=1+y^{2},\) and below by the \(x y\) -plane.

Sketch the region of integration and the solid whose volume is given by the double integral. $$\int_{0}^{3} \int_{0}^{2-2 x / 3}\left(1-\frac{1}{3} x-\frac{1}{2} y\right) d y d x$$

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 \(y=0\).

Sketch the region of integration and the solid whose volume is given by the double integral. $$\int_{0}^{4} \int_{-\sqrt{16-y^{2}}}^{\sqrt{16-y^{2}}} \sqrt{25-x^{2}-y^{2}} d x d y$$

Improper double integrals can often be computed similarly to improper integrals of one variable. The first iteration of the following improper integrals is conducted just as if they were proper integrals. One then evaluates an improper integral of a single variable by taking appropriate limits, as in Section \(8.7 .\) Evaluate the improper integrals as iterated integrals. $$\int_{1}^{\infty} \int_{e^{-}}^{1} \frac{1}{x^{3} y} d y d x$$

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

Improper double integrals can often be computed similarly to improper integrals of one variable. The first iteration of the following improper integrals is conducted just as if they were proper integrals. One then evaluates an improper integral of a single variable by taking appropriate limits, as in Section \(8.7 .\) Evaluate the improper integrals as iterated integrals. $$\int_{-1}^{1} \int_{-1 / \sqrt{1-x^{2}}}^{1 / \sqrt{1-x^{2}}}(2 y+1) d y d x$$

Improper double integrals can often be computed similarly to improper integrals of one variable. The first iteration of the following improper integrals is conducted just as if they were proper integrals. One then evaluates an improper integral of a single variable by taking appropriate limits, as in Section \(8.7 .\) Evaluate the improper integrals as iterated integrals. $$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{1}{\left(x^{2}+1\right)\left(y^{2}+1\right)} d x d y$$

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 \(\theta=\pi / 3\) \(r \geq 0\).

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 \(\delta=1\).)

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.)

Circular sector Integrate \(f(x, y)=\sqrt{4-x^{2}}\) over the smaller sector cut from the disk \(x^{2}+y^{2} \leq 4\) by the rays \(\theta=\pi / 6\) and \(\theta=\pi / 2\)

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 a. \(\delta(r, \theta, z)=z\) b. \(\delta(r, \theta, z)=r\)

Unboundedregion Integrate \(f(x, y)=1 /\left[\left(x^{2}-x\right)(y-1)^{2 / 3}\right]\) over the infinite rectangle \(2 \leq x<\infty, 0 \leq y \leq 2\)

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 a. \(\delta(r, \theta, z)=z\) b. \(\delta(r, \theta, z)=z^{2}\)

Noncircular cylinder \(\quad\) A solid right (noncircular) cylinder has its base \(R\) in the \(x y\) -plane and is bounded above by the paraboloid \(z=x^{2}+y^{2} .\) The cylinder's volume is $$V=\int_{0}^{1} \int_{0}^{y}\left(x^{2}+y^{2}\right) d x d y+\int_{1}^{2} \int_{0}^{2-y}\left(x^{2}+y^{2}\right) d x d y$$ Sketch the base region \(R\) and express the cylinder's volume as a single iterated integral with the order of integration reversed. Then evaluate the integral to find the volume.

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

Converting to a double integral Evaluate the integral $$\int_{0}^{2}\left(\tan ^{-1} \pi x-\tan ^{-1} x\right) d x$$ (Hint: Write the integrand as an integral.)

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.

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.)

Minimizing a double integral What region \(R\) in the \(x y\) -plane minimizes the value of $$\iint_{R}\left(x^{2}+y^{2}-9\right) d A ?$$ Give reasons for your answer.

Is it possible to evaluate the integral of a continuous function \(f(x, y)\) over a rectangular region in the \(x y\) -plane and get different answers depending on the order of integration? Give reasons for your answer.

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.

How would you evaluate the double integral of a continuous function \(f(x, y)\) over the region \(R\) in the \(x y\) -plane enclosed by the triangle with vertices \((0,1),(2,0),\) and (1,2)\(?\) Give reasons for your answer.

Unbounded region Prove that $$\begin{aligned} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-x^{2}-y^{2}} d x d y &=\lim _{b \rightarrow \infty} \int_{-b}^{b} \int_{-b}^{b} e^{-x^{2}-y^{2}} d x d y \\ &=4\left(\int_{0}^{\infty} e^{-x^{2}} d x\right)^{2} \end{aligned}$$

Improper double integral Evaluate the improper integral $$\int_{0}^{1} \int_{0}^{3} \frac{x^{2}}{(y-1)^{2 / 3}} d y d x$$

Use a CAS double-integral evaluator to estimate the values of the integrals. $$\int_{1}^{3} \int_{1}^{x} \frac{1}{x y} d y d x$$