Chapter 13

In Problems 21-32, sketch the indicated solid. Then find its volume by an iterated integration. Solid in the first octant bounded by the surface \(9 z=36-9 x^{2}-4 y^{2}\) 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 between \(z=x^{2}+y^{2}+2\) and \(z=1\) and lying above \(R=\\{(x, y):-1 \leq x \leq 1,0 \leq y \leq 1\\}\)

In Problems 29-32, write the given iterated integral as an iterated integral with the indicated order of integration. $$ \int_{0}^{2} \int_{0}^{9-x^{2}} \int_{0}^{2-x} f(x, y, z) d z d y d x ; d y d x d z $$

The depth (in feet) of water distributed by a rotating lawn sprinkler in an hour is \(k e^{-r / 10}, 0 \leq r \leq 10\), where \(r\) is the distance from the sprinkler and \(k\) is a constant. Determine \(k\) if 100 cubic feet of water is distributed in 1 hour.

In Problems 21-32, 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

Find the volume of the given solid. First, sketch the solid; then estimate its volume; finally, determine its exact volume Solid in the first octant enclosed by \(z=4-x^{2}\) and \(y=2\)

In Problems 29-32, write the given iterated integral as an iterated integral with the indicated order of integration. $$ \int_{0}^{2} \int_{0}^{9-x^{2}} \int_{0}^{2-x} f(x, y, z) d z d y d x ; d z d x d y $$

Show that if \(f(x, y)=g(x) h(y)\) then $$ \int_{a}^{b} \int_{c}^{d} f(x, y) d y d x=\left[\int_{a}^{b} g(x) d x\right]\left[\int_{c}^{d} h(y) d y\right] $$

Let \(S_{1}\) and \(S_{2}\) be disjoint laminas in the \(x y\)-plane of mass \(m_{1}\) and \(m_{2}\) with centers of mass \(\left(\bar{x}_{1}, \bar{y}_{1}\right)\) and \(\left(\bar{x}_{2}, \bar{y}_{2}\right)\). Show that the center of mass \((\bar{x}, \bar{y})\) of the combined lamina \(S_{1} \cup S_{2}\) satisfies $$ \bar{x}=\bar{x}_{1} \frac{m_{1}}{m_{1}+m_{2}}+\bar{x}_{2} \frac{m_{2}}{m_{1}+m_{2}} $$ with a similar formula for \(\bar{y}\). Conclude that in finding \((\bar{x}, \bar{y})\) the two laminas can be treated as if they were point masses at \(\left(\bar{x}_{1}, \bar{y}_{1}\right)\) and \(\left(\bar{x}_{2}, \bar{y}_{2}\right)\).

Consider the ring \(A\) of height \(2 b\) obtained from a sphere of radius \(a\) when a hole of radius \(c(c

Evaluate $$ \int_{0}^{1} \int_{0}^{1} x y e^{x^{2}+y^{2}} d y d x $$

Let \(S\) be a lamina in the \(x y\)-plane with center of mass at the origin, and let \(L\) be the line \(a x+b y=0\), which goes through the origin. Show that the (signed) distance \(d\) of a point \((x, y)\) from \(L\) is \(d=(a x+b y) / \sqrt{a^{2}+b^{2}}\), and use this to conclude that the moment of \(S\) with respect to \(L\) is 0 . Note: This shows that a lamina will balance on any line through its center of mass.

Find the volume of the solid trapped between the surface \(z=\cos x \cos y\) and the \(x y\)-plane, where \(-\pi \leq x \leq \pi\), \(-\pi \leq y \leq \pi\).

Show that $$ \int_{0}^{\infty} \int_{0}^{\infty} \frac{1}{\left(1+x^{2}+y^{2}\right)^{2}} d y d x=\frac{\pi}{4} $$

In Problems 33-38, write the given iterated integral as an iterated integral with the order of integration interchanged. Hint: Begin by sketching a region \(S\) and representing it in two ways, as in Example \(5 .\) \(\int_{0}^{1} \int_{-y}^{y} f(x, y) d x d y\)

Evaluate each iterated integral. \(\int_{-2}^{2} \int_{-1}^{1}\left|x^{2} y^{3}\right| d y d x\)

Recall the formula \(A=\frac{1}{2} r^{2} \theta\) for the area of the sector of a circle of radius \(r\) and central angle \(\theta\) radians (Section 10.7). Use this to obtain the formula $$ A=\frac{r_{1}+r_{2}}{2}\left(r_{2}-r_{1}\right)\left(\theta_{2}-\theta_{1}\right) $$ for the area of the polar rectangle \(\left\\{(r, \theta): r_{1} \leq r \leq r_{2}\right.\), \(\left.\theta_{1} \leq \theta \leq \theta_{2}\right\\} .\)

In Problems 33-38, write the given iterated integral as an iterated integral with the order of integration interchanged. Hint: Begin by sketching a region \(S\) and representing it in two ways, as in Example \(5 .\) \(\int_{-1}^{0} \int_{-\sqrt{y+1}}^{\sqrt{y+1}} f(x, y) d x d y\)

Evaluate each iterated integral. \(\int_{-2}^{2} \int_{-1}^{1}\left[x^{2}\right]\left|y^{3}\right| d y d x\)

Prove the Cauchy-Schwarz Inequality for Integrals: \(\left[\int_{a}^{b} f(x) g(x) d x\right]^{2} \leq \int_{a}^{b} f^{2}(x) d x \int_{a}^{b} g^{2}(x) d x\) Hint: Consider the double integral of $$ F(x, y)=[f(x) g(y)-f(y) g(x)]^{2} $$ over the rectangle \(R=\\{(x, y): a \leq x \leq b, a \leq y \leq b\\}\).

Evaluate \(\iint_{S} \sin \left(x y^{2}\right) d A\), where \(S\) is the annulus \(\left\\{(x, y): 1 \leq x^{2}+y^{2} \leq 4\right\\}\). Hint: Done without thinking, this problem is hard; using symmetry, it is trivial.

Evaluate \(\iint_{S} \sin \left(y^{3}\right) d A\), where \(S\) is the region bounded by \(y=\sqrt{x}, y=2\), and \(x=0\). Hint: If one order of integration does not work, try the other.

Suppose that the random variables \((X, Y)\) have joint PDF \(f(x, y)= \begin{cases}k y, & \text { if } 0 \leq x \leq 12 ; 0 \leq y \leq x \\ 0, & \text { otherwise }\end{cases}\) Find each of the following: (a) \(k\) (b) \(P(Y>4)\) (c) \(E(X)\)

Evaluate \(\iint_{S} x^{2} d A\), where \(S\) is the region between the ellipse \(x^{2}+2 y^{2}=4\) and the circle \(x^{2}+y^{2}=4\).

Suppose that the random variables \((X, Y, Z)\) have joint PDF $$ f(x, y, z)= \begin{cases}k x y, & \text { if } 0 \leq x \leq y ; 0 \leq y \leq 4 ; 0 \leq z \leq 2 \\ 0, & \text { otherwise }\end{cases} $$ Find each of the following: (a) \(k\) (b) \(P(X>2)\) (c) \(E(X)\)

Suppose that the random variables \((X, Y)\) have joint PDF $$ f(x, y)= \begin{cases}\frac{3}{256}\left(x^{2}+y^{2}\right), & \text { if } 0 \leq x \leq y ; 0 \leq y \leq 4 \\ 0, & \text { otherwise }\end{cases} $$ Find each of the following: (a) \(P(X>2)\) (b) \(P(X+Y \leq 4)\) (c) \(E(X+Y)\)