Chapter 14

Evaluate the integrals in Exercises \(7-20\). $$\int_{0}^{\pi} \int_{0}^{\pi} \int_{0}^{\pi} \cos (u+v+w) d u d v d w \quad(u v w-\text { space })$$

Find the center of mass and the moment of inertia about the \(y\) -axis of a thin plate bounded by the \(x\) -axis, the lines \(x=\pm 1,\) and the parabola \(y=x^{2}\) if \(\delta(x, y)=7 y+1\).

Write an iterated integral for \(\iint_{R} d A\) over the described region \(R\) using (a) vertical cross-sections, (b) horizontal cross-sections. Bounded by \(y=3-2 x, y=x,\) and \(x=0\)

The integrals and sums of integrals in Exercises \(13-18\) give the areas of regions in the \(x y\) -plane. Sketch each region, label each bounding curve with its equation, and give the coordinates of the points where the curves intersect. Then find the area of the region. $$\int_{-1}^{0} \int_{-2 x}^{1-x} d y d x+\int_{0}^{2} \int_{-x / 2}^{1-x} d y d x$$

Evaluate the double integral over the given region \(R\). $$\iint_{R} x y \cos y d A, \quad R: \quad-1 \leq x \leq 1, \quad 0 \leq y \leq \pi$$

Change the Cartesian integral into an equivalent polar integral. Then evaluate the polar integral. $$\int_{-1}^{1} \int_{-\sqrt{1-x^{2}}}^{\sqrt{1-x^{2}}} \frac{2}{\left(1+x^{2}+y^{2}\right)^{2}} d y d x$$

Volume of an ellipsoid Find the volume of the ellipsoid $$ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1 $$ (Hint: Let \(x=a u, y=b v,\) and \(z=c w .\) Then find the volume of an appropriate region in \(u v w\) -space.)

Evaluate the integrals in Exercises \(7-20\). $$\int_{0}^{1} \int_{1}^{\sqrt{e}} \int_{1}^{e} s e^{s} \ln r \frac{(\ln t)^{2}}{t} d t d r d s \quad(r s t-\text { space })$$

Find the center of mass and the moment of inertia about the \(x\) -axis of a thin rectangular plate bounded by the lines \(x=0, x=20, y=-1,\) and \(y=1\) if \(\delta(x, y)=1+(x / 20)\).

Write an iterated integral for \(\iint_{R} d A\) over the described region \(R\) using (a) vertical cross-sections, (b) horizontal cross-sections. Bounded by \(y=x^{2}\) and \(y=x+2\)

The integrals and sums of integrals in Exercises \(13-18\) give the areas of regions in the \(x y\) -plane. Sketch each region, label each bounding curve with its equation, and give the coordinates of the points where the curves intersect. Then find the area of the region. $$\int_{0}^{2} \int_{x^{2}-4}^{0} d y d x+\int_{0}^{4} \int_{0}^{\sqrt{x}} d y d x$$

Evaluate the double integral over the given region \(R\). $$\iint_{R} y \sin (x+y) d A, \quad R: \quad-\pi \leq x \leq 0, \quad 0 \leq y \leq \pi$$

Evaluate the integrals in Exercises \(7-20\). $$\int_{0}^{\pi / 4} \int_{0}^{\ln \sec v} \int_{-\infty}^{2 t} e^{x} d x d t d v \quad(t v x-\text { space })$$

Find the center of mass, the moment of inertia about the coordinate axes, and the polar moment of inertia of a thin triangular plate bounded by the lines \(y=x, y=-x,\) and \(y=1\) if \(\delta(x, y)=y+1\).

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

Find the average value of \(f(x, y)=\sin (x+y)\) over a. the rectangle \(0 \leq x \leq \pi, \quad 0 \leq y \leq \pi\) b. the rectangle \(0 \leq x \leq \pi, \quad 0 \leq y \leq \pi / 2\)

Evaluate the double integral over the given region \(R\). $$\iint_{R} e^{x-y} d A, \quad R: \quad 0 \leq x \leq \ln 2, \quad 0 \leq y \leq \ln 2$$

Change the Cartesian integral into an equivalent polar integral. Then evaluate the polar integral. $$\int_{-1}^{1} \int_{-\sqrt{1-y^{2}}}^{\sqrt{1-y^{2}}} \ln \left(x^{2}+y^{2}+1\right) d x d y$$

Evaluate the integrals in Exercises \(7-20\). $$\int_{0}^{7} \int_{0}^{2} \int_{0}^{\sqrt{4-q^{2}}} \frac{q}{r+1} d p d q d r \quad(\text {pqr-space})$$

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

Which do you think will be larger, the average value of \(f(x, y)=x y\) over the square \(0 \leq x \leq 1,0 \leq y \leq 1,\) or the average value of \(f\) over the quarter circle \(x^{2}+y^{2} \leq 1\) in the first quadrant? Calculate them to find out.

Change the Cartesian integral into an equivalent polar integral. Then evaluate the polar integral. $$\int_{0}^{1} \int_{x}^{\sqrt{2-x^{2}}}(x+2 y) d y d x$$

Evaluate the spherical coordinate integrals. $$\int_{0}^{\pi} \int_{0}^{\pi} \int_{0}^{2 \sin \phi} \rho^{2} \sin \phi d \rho d \phi d \theta$$

Find the Jacobian \(\partial(x, y) / \partial(u, v)\) of the transformation a. \(x=u \cos v, \quad y=u \sin v\) b. \(x=u \sin v, \quad y=u \cos v\)

Find the average height of the paraboloid \(z=x^{2}+y^{2}\) over the square \(0 \leq x \leq 2,0 \leq y \leq 2\).

Sketch the region of integration and evaluate the integral. $$\int_{1}^{\ln 8} \int_{0}^{\ln y} e^{x+y} d x d y$$

Evaluate the double integral over the given region \(R\). $$\iint_{R} \frac{x y^{3}}{x^{2}+1} d A, \quad R: \quad 0 \leq x \leq 1, \quad 0 \leq y \leq 2$$

Change the Cartesian integral into an equivalent polar integral. Then evaluate the polar integral. $$\int_{1}^{2} \int_{0}^{\sqrt{2 x-x^{2}}} \frac{1}{\left(x^{2}+y^{2}\right)^{2}} d y d x$$

Evaluate the spherical coordinate integrals. $$\int_{0}^{2 \pi} \int_{0}^{\pi / 4} \int_{0}^{2}(\rho \cos \phi) \rho^{2} \sin \phi d \rho d \phi d \theta$$

Find the Jacobian \(\partial(x, y, z) / \partial(u, v, w)\) of the transformation a. \(x=u \cos v, \quad y=u \sin v, \quad z=w\) b. \(x=2 u-1, \quad y=3 v-4, \quad z=(1 / 2)(w-4)\)

Find the average value of \(f(x, y)=1 /(x y)\) over the square \(\ln 2 \leq x \leq 2 \ln 2, \ln 2 \leq y \leq 2 \ln 2\).

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

Evaluate the double integral over the given region \(R\). $$\iint_{R} \frac{y}{x^{2} y^{2}+1} d A, \quad R: \quad 0 \leq x \leq 1, \quad 0 \leq y \leq 1$$

Find the volumes of the regions in Exercises \(23-36\). The region between the cylinder \(z=y^{2}\) and the \(x y\) -plane that is bounded by the planes \(x=0, x=1, y=-1, y=1\) (GRAPH CAN'T COPY).

Sketch the region of integration and convert each polar integral or sum of integrals to a Cartesian integral or sum of integrals. Do not evaluate the integrals. $$\int_{0}^{\pi / 2} \int_{0}^{1} r^{3} \sin \theta \cos \theta d r d \theta$$

Evaluate the spherical coordinate integrals. $$\int_{0}^{2 \pi} \int_{0}^{\pi} \int_{0}^{(1-\cos \phi) / 2} \rho^{2} \sin \phi d \rho d \phi d \theta$$

A solid "trough" of constant density is bounded below by the surface \(z=4 y^{2},\) above by the plane \(z=4,\) and on the ends by the planes \(x=1\) and \(x=-1 .\) Find the center of mass and the moments of inertia with respect to the three axes.

Geometric area Find the area of the region $$ R: 0 \leq x \leq 2,2-x \leq y \leq \sqrt{4-x^{2}} $$ using (a) Fubini's Theorem, (b) simple geometry.

Sketch the region of integration and evaluate the integral. $$\int_{0}^{1} \int_{0}^{y^{2}} 3 y^{3} e^{r y} d x d y$$

Integrate \(f\) over the given region. Square \(\quad f(x, y)=1 /(x y)\) over the square \(1 \leq x \leq 2\) \(1 \leq y \leq 2\)

Sketch the region of integration and convert each polar integral or sum of integrals to a Cartesian integral or sum of integrals. Do not evaluate the integrals. $$\int_{\pi / 6}^{\pi / 2} \int_{1}^{\csc \theta} r^{2} \cos \theta d r d \theta$$

Evaluate the spherical coordinate integrals. $$\int_{0}^{3 \pi / 2} \int_{0}^{\pi} \int_{0}^{1} 5 \rho^{3} \sin ^{3} \phi d \rho d \phi d \theta$$

Geometric area Find the area of the circular washer with outer radius 2 and inner radius \(1,\) using (a) Fubini's Theorem, (b) simple geometry.

Sketch the region of integration and evaluate the integral. $$\int_{1}^{4} \int_{0}^{\sqrt{x}} \frac{3}{2} e^{y / \sqrt{x}} d y d x$$

Integrate \(f\) over the given region. Rectangle \(\quad f(x, y)=y \cos x y\) over the rectangle \(0 \leq x \leq \pi\) \(0 \leq y \leq 1\)

Sketch the region of integration and convert each polar integral or sum of integrals to a Cartesian integral or sum of integrals. Do not evaluate the integrals. $$\int_{0}^{\pi / 4} \int_{0}^{2 \sec \theta} r^{5} \sin ^{2} \theta d r d \theta$$

Evaluate the spherical coordinate integrals. $$\int_{0}^{2 \pi} \int_{0}^{\pi / 3} \int_{\sec \phi}^{2} 3 \rho^{2} \sin \phi d \rho d \phi d \theta$$

Centroid of a solid semiellipsoid Assuming the result that the centroid of a solid hemisphere lies on the axis of symmetry threeeighths of the way from the base toward the top, show, by transforming the appropriate integrals, that the center of mass of a solid semiellipsoid \(\left(x^{2} / a^{2}\right)+\left(y^{2} / b^{2}\right)+\left(z^{2} / c^{2}\right) \leq 1, z \geq 0\) lies on the \(z\) -axis three-eighths of the way from the base toward the top. (You can do this without evaluating any of the integrals.)

a. Find the center of mass of a solid of constant density bounded below by the paraboloid \(z=x^{2}+y^{2}\) and above by the plane \(z=4\) b. Find the plane \(z=c\) that divides the solid into two parts of equal volume. This plane does not pass through the center of mass.

Integrate \(f\) over the given region. Quadrilateral \(f(x, y)=x / y\) over the region in the first quadrant bounded by the lines \(y=x, y=2 x, x=1,\) and \(x=2\)