Chapter 14

Sketch the region of integration and write an equivalent double integral with the order of integration reversed. $$\int_{0}^{1} \int_{1}^{e^{x}} d y d x$$

In Exercises \(37-40,\) find the average value of \(F(x, y, z)\) over the given region. \(F(x, y, z)=x+y-z\) over the rectangular solid in the first octant bounded by the coordinate planes and the planes \(x=1, y=1,\) and \(z=2\).

Converting to a polar integral Integrate \(f(x, y)=\) \(\left[\ln \left(x^{2}+y^{2}\right)\right] /\left(x^{2}+y^{2}\right)\) over the region \(1 \leq x^{2}+y^{2} \leq e^{2}.\)

Sketch the region of integration and write an equivalent double integral with the order of integration reversed. $$\int_{0}^{\ln 2} \int_{e^{y}}^{2} d x d y$$

In Exercises \(37-40,\) find the average value of \(F(x, y, z)\) over the given region. \(F(x, y, z)=x^{2}+y^{2}+z^{2}\) over the cube in the first octant bounded by the coordinate planes and the planes \(x=1, y=1\) and \(z=1\).

The region that lies inside the cardioid \(r=1+\cos \theta\) and outside the circle \(r=1\) is the base of a solid right cylinder. The top of the cylinder lies in the plane \(z=x .\) Find the cylinder's volume.

Set up triple integrals for the volume of the sphere \(\rho=2\) in (a) spherical, (b) cylindrical, and (c) rectangular coordinates.

Sketch the region of integration and write an equivalent double integral with the order of integration reversed. $$\int_{0}^{3 / 2} \int_{0}^{9-4 x^{2}} 16 x d y d x$$

In Exercises \(37-40,\) find the average value of \(F(x, y, z)\) over the given region. \(F(x, y, z)=x y z\) over the cube in the first octant bounded by the coordinate planes and the planes \(x=2, y=2,\) and \(z=2\).

The region enclosed by the lemniscate \(r^{2}=2 \cos 2 \theta\) is the base of a solid right cylinder whose top is bounded by the sphere \(z=\sqrt{2-r^{2}} .\) Find the cylinder's volume.

Let \(D\) be the region in the first octant that is bounded below by the cone \(\phi=\pi / 4\) and above by the sphere \(\rho=3 .\) Express the volume of \(D\) as an iterated triple integral in (a) cylindrical and (b) spherical coordinates. Then (c) find \(V\).

Sketch the region of integration and write an equivalent double integral with the order of integration reversed. $$\int_{0}^{2} \int_{0}^{4-y^{2}} y d x d y$$

a. The usual way to evaluate the improper integral \(I=\int_{0}^{\infty} e^{-x^{2}} d x\) is first to calculate its square: $$I^{2}=\left(\int_{0}^{\infty} e^{-x^{2}} d x\right)\left(\int_{0}^{\infty} e^{-y^{2}} d y\right)=\int_{0}^{\infty} \int_{0}^{\infty} e^{-\left(x^{2}+y^{2}\right)} d x d y.$$ Evaluate the last integral using polar coordinates and solve the resulting equation for \(I.\) b. Evaluate $$\lim _{x \rightarrow \infty} \operatorname{erf}(x)=\lim _{x \rightarrow \infty} \int_{0}^{x} \frac{2 e^{-t^{2}}}{\sqrt{\pi}} d t.$$

Let \(D\) be the smaller cap cut from a solid ball of radius 2 units by a plane 1 unit from the center of the sphere. Express the volume of \(D\) as an iterated triple integral in (a) spherical, (b) cylindrical, and (c) rectangular coordinates. Then (d) find the volume by evaluating one of the three triple integrals.

Sketch the region of integration and write an equivalent double integral with the order of integration reversed. $$\int_{0}^{1} \int_{-\sqrt{1-y^{2}}}^{\sqrt{1-y^{2}}} 3 y d x d y$$

Evaluate the integral $$\int_{0}^{\infty} \int_{0}^{\infty} \frac{1}{\left(1+x^{2}+y^{2}\right)^{2}} d x d y$$

Express the moment of inertia \(I_{z}\) of the solid hemisphere \(x^{2}+y^{2}+z^{2} \leq 1, z \geq 0,\) as an iterated integral in (a) cylindrical and (b) spherical coordinates. Then (c) find \(I_{z}\).

Sketch the region of integration and write an equivalent double integral with the order of integration reversed. $$\int_{0}^{2} \int_{-\sqrt{4-x^{2}}}^{\sqrt{4-x^{2}}} 6 x d y d x$$

Integrate the function \(f(x, y)=1 /\left(1-x^{2}-y^{2}\right)\) over the disk \(x^{2}+y^{2} \leq 3 / 4 .\) Does the integral of \(f(x, y)\) over the disk \(x^{2}+y^{2} \leq 1\) exist? Give reasons for your answer.

Sketch the region of integration and write an equivalent double integral with the order of integration reversed. $$\int_{1}^{e} \int_{0}^{\ln x} x y d y d x$$

Evaluate the integrals in Exercises \(41-44\) by changing the order of integration in an appropriate way. $$\int_{0}^{2} \int_{0}^{4-x^{2}} \int_{0}^{x} \frac{\sin 2 z}{4-z} d y d z d x$$

Use the double integral in polar coordinates to derive the formula $$A=\int_{\alpha}^{\beta} \frac{1}{2} r^{2} d \theta$$ for the area of the fan-shaped region between the origin and polar curve \(r=f(\theta), \alpha \leq \theta \leq \beta\).

Sketch the region of integration and write an equivalent double integral with the order of integration reversed. $$\int_{0}^{\pi / 6} \int_{\sin x}^{1 / 2} x y^{2} d y d x$$

Solve for \(a:\) $$\int_{0}^{1} \int_{0}^{4-a-x^{2}} \int_{a}^{4-x^{2}-y} d z d y d x=\frac{4}{15}$$

Let \(P_{0}\) be a point inside a circle of radius \(a\) and let \(h\) denote the distance from \(P_{0}\) to the center of the circle. Let \(d\) denote the distance from an arbitrary point \(P\) to \(P_{0} .\) Find the average value of \(d^{2}\) over the region enclosed by the circle. (Hint: Simplify your work by placing the center of the circle at the origin and \(P_{0}\) on the \(x\)-axis.)

Sketch the region of integration and write an equivalent double integral with the order of integration reversed. $$\int_{0}^{3} \int_{1}^{e^{y}}(x+y) d x d y$$

For what value of \(c\) is the volume of the ellipsoid \(x^{2}+(y / 2)^{2}+(z / c)^{2}=1\) equal to \(8 \pi ?\)

Suppose that the area of a region in the polar coordinate plane is $$A=\int_{\pi / 4}^{3 \pi / 4} \int_{\csc \theta}^{2 \sin \theta} r d r d \theta$$ Sketch the region and find its area.

Sketch the region of integration and write an equivalent double integral with the order of integration reversed. $$\int_{0}^{\sqrt{3}} \int_{0}^{\tan ^{-1} y} \sqrt{x y} d x d y$$

What domain \(D\) in space minimizes the value of the integral $$\iiint_{D}\left(4 x^{2}+4 y^{2}+z^{2}-4\right) d V ?$$ Give reasons for your answer.

Evaluate the integral \(\iint_{R} \sqrt{x^{2}+y^{2}} d A,\) where \(R\) is the region inside the upper semicircle of radius 2 centered at the origin, but outside the circle \(x^{2}+(y-1)^{2}=1\).

Sketch the region of integration, reverse the order of integration, and evaluate the integral. $$\int_{0}^{\pi} \int_{x}^{\pi} \frac{\sin y}{y} d y d x$$

What domain \(D\) in space maximizes the value of the integral $$\iiint_{D}\left(1-x^{2}-y^{2}-z^{2}\right) d V ?$$ Give reasons for your answer.

Evaluate the integral \(\iint_{R}\left(x^{2}+y^{2}\right)^{-2} d A,\) where \(R\) is the region inside the circle \(x^{2}+y^{2}=2\) for \(x \leq-1.\)

Sketch the region of integration, reverse the order of integration, and evaluate the integral. $$\int_{0}^{2} \int_{x}^{2} 2 y^{2} \sin x y d y d x$$

Use a CAS to change the Cartesian integrals into an equivalent polar integral and evaluate the polar integral. Perform the following steps in each exercise. a. Plot the Cartesian region of integration in the \(x y\)-plane. b. Change each boundary curve of the Cartesian region in part (a) to its polar representation by solving its Cartesian equation for \(r\) and \(\theta\) c. Using the results in part (b), plot the polar region of integration in the \(r \theta\)-plane. d. Change the integrand from Cartesian to polar coordinates. Determine the limits of integration from your plot in part (c) and evaluate the polar integral using the CAS integration utility. $$\int_{0}^{1} \int_{x}^{1} \frac{y}{x^{2}+y^{2}} d y d x$$

In Exercises \(49-52,\) use a CAS integration utility to evaluate the triple integral of the given function over the specified solid region. \(F(x, y, z)=x^{2} y^{2} z\) over the solid cylinder bounded by \(x^{2}+y^{2}=1\) and the planes \(z=0\) and \(z=1\).

Find the volume of the portion of the solid sphere \(\rho \leq a\) that lies between the cones \(\phi=\pi / 3\) and \(\phi=2 \pi / 3\).

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

Use a CAS to change the Cartesian integrals into an equivalent polar integral and evaluate the polar integral. Perform the following steps in each exercise. a. Plot the Cartesian region of integration in the \(x y\)-plane. b. Change each boundary curve of the Cartesian region in part (a) to its polar representation by solving its Cartesian equation for \(r\) and \(\theta\) c. Using the results in part (b), plot the polar region of integration in the \(r \theta\)-plane. d. Change the integrand from Cartesian to polar coordinates. Determine the limits of integration from your plot in part (c) and evaluate the polar integral using the CAS integration utility. $$\int_{0}^{1} \int_{0}^{x / 2} \frac{x}{x^{2}+y^{2}} d y d x$$

In Exercises \(49-52,\) use a CAS integration utility to evaluate the triple integral of the given function over the specified solid region. \(F(x, y, z)=|x y z|\) over the solid bounded below by the paraboloid \(z=x^{2}+y^{2}\) and above by the plane \(z=1\).

Find the volume of the region cut from the solid sphere \(\rho \leq a\) by the half-planes \(\theta=0\) and \(\theta=\pi / 6\) in the first octant.

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

Find the volume of the smaller region cut from the solid sphere \(\rho \leq 2\) by the plane \(z=1\).

Sketch the region of integration, reverse the order of integration, and evaluate the integral. $$\int_{0}^{2 \sqrt{\ln 3}} \int_{y / 2}^{\sqrt{\ln 3}} e^{x^{2}} d x d y$$

In Exercises \(49-52,\) use a CAS integration utility to evaluate the triple integral of the given function over the specified solid region. \(F(x, y, z)=x^{4}+y^{2}+z^{2}\) over the solid sphere \(x^{2}+y^{2}+\) \(z^{2} \leq 1\).

Find the volume of the solid enclosed by the cone \(z=\sqrt{x^{2}+y^{2}}\) between the planes \(z=1\) and \(z=2\).

Sketch the region of integration, reverse the order of integration, and evaluate the integral. $$\int_{0}^{3} \int_{\sqrt{x / 3}}^{1} e^{y^{3}} d y d x$$

Find the volume of the region bounded below by the plane \(z=0,\) laterally by the cylinder \(x^{2}+y^{2}=1,\) and above by the paraboloid \(z=x^{2}+y^{2}\).

Sketch the region of integration, reverse the order of integration, and evaluate the integral. $$\int_{0}^{1 / 16} \int_{y^{1 / 4}}^{1 / 2} \cos \left(16 \pi x^{5}\right) d x d y$$