Chapter 13

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 \(R=\\{(x, y):-1 \leq x \leq 1,0 \leq y \leq 1\\}\)

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

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.

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 \(y=2\)

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 \(\left(\bar{x}_{2}, \bar{y}_{2}\right)\).

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 \(5 .\) $$ \int_{0}^{1} \int_{0}^{x} f(x, y) d y d x $$

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

Find the volume of the wedge cut from a tall right circular cylinder of radius \(a\) by a plane through a diameter of its base and making an angle \(\alpha(0<\alpha<\pi / 2)\) with the base (compare Problem 39, Section 6.2).

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 \(5 .\) $$ \int_{0}^{2} \int_{y^{2}}^{2 y} f(x, y) d x d y $$

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.

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 \(5 .\) $$ \int_{0}^{1} \int_{x^{2}}^{x^{1 / 4}} f(x, y) d y d x $$

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

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

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 \(5 .\) $$ \int_{1 / 2}^{1} \int_{x^{3}}^{x} f(x, y) d y d x $$

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

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

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 \(5 .\) $$ \int_{0}^{1} \int_{-y}^{y} f(x, y) d x d y $$

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 \(\left\\{(r, \theta): r_{1} \leq r \leq r_{2},\right.\), \(\left.\theta_{1} \leq \theta \leq \theta_{2}\right\\}\).

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 \(\int_{0}^{\sqrt{3}} \int_{0}^{1} \frac{8 x}{\left(x^{2}+y^{2}+1\right)^{2}} d y d x .\) Hint: Reverse the order of integration.

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.

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

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 \((X, Y, Z)\) have joint PDF $$ f(x, y, z)=\left\\{\begin{array}{ll} k x y, & \text { if } 0 \leq x \leq y ; 0 \leq y \leq 4 ; 0 \leq z \leq 2 \\ 0, & \text { otherwise } \end{array}\right. $$ Find each of the following: (a) \(k\) (b) \(P(X>2)\) (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 \((X, Y)\) have joint \(\mathrm{PDF}\) \(f(x, y)=\left\\{\begin{array}{ll}\frac{3}{256}\left(x^{2}+y^{2}\right), & \text { if } 0 \leq x \leq y ; 0 \leq y \leq 4 \\ 0, & \text { otherwise }\end{array}\right.\) Find each of the following: (a) \(P(X>2)\) (b) \(P(X+Y \leq 4)\) (c) \(E(X+Y)\)

Suppose that \(f(x, y)\) is a continuous function defined on a region \(R\) that is closed and bounded. Show that there is an ordered pair \((a, b)\) in \(R\) such that $$ \iint_{R} f(x, y) d A=f(a, b) A(R) $$ This result is called the Mean Value Theorem for Double Integrals. Hint: You will need the Intermediate Value Theorem (Theorem 2.7F).