Chapter 6

Find the volume of the solid generated when the region in the first quadrant bounded above by \(y=2\) and on the right by \(y=-\ln x\) is revolved about the \(y\) -axis.

Let \(0 \leq f(x) \leq g(x)\) for all \(x\) in \([0,1]\), and let \(R\) and \(S\) be the regions under the graphs of \(f\) and \(g\), respectively. Prove of disprove that \(\bar{y}_{R} \leq \bar{y}_{S}\).

Suppose a random variable \(Y\) has CDF $$ F(y)=\left\\{\begin{array}{ll} 0, & \text { if } y<0 \\ 2 y /(y+1), & \text { if } 0 \leq y \leq 1 \\ 1, & \text { if } y>1 \end{array}\right. $$ Find each of the following: (a) \(P(Y<2)\) (b) \(P(0.5

Find the volume of the solid generated by revolving the region in the first quadrant bounded by the curve \(y^{2}=x^{3}\), the line \(x=4\), and the \(x\) -axis: (a) about the line \(x=4\); (b) about the line \(y=8\).

Starting at \(s=0\) when \(t=0\), an object moves along a line so that its velocity at time \(t\) is \(v(t)=2 t-4\) centimeters per second. How long will it take to get to \(s=12 ?\) To travel a total distance of 12 centimeters?

Suppose a random variable \(Z\) has CDF $$ F(z)=\left\\{\begin{array}{ll} 0, & \text { if } z<0 \\ z^{2} / 9, & \text { if } 0 \leq z \leq 3 \\ 1, & \text { if } z>3 \end{array}\right. $$ Find each of the following: (a) \(P(Z>1)\) (b) \(P(1

Show that the area of the part of the surface of a sphere of radius \(a\) between two parallel planes \(h\) units apart \((h<2 a)\) is \(2 \pi a h .\) Thus, show that if a right circular cylinder is circumscribed about a sphere then two planes parallel to the base of the cylinder bound regions of the same area on the sphere and the cylinder.

Find the volume of the solid generated by revolving the region bounded by the curve \(y^{2}=x^{3}\), the line \(y=8\), and the \(y\) -axis: (a) about the line \(x=4\); (b) about the line \(y=8\).

Consider the curve \(y=1 / x^{2}\) for \(1 \leq x \leq 6\). (a) Calculate the area under this curve. (b) Determine \(c\) so that the line \(x=c\) bisects the area of part (a). (c) Determine \(d\) so that the line \(y=d\) bisects the area of nart (a).

The expected value of a function \(g(X)\) of a continuous random variable \(X\) having \(\operatorname{PDF} f(x)\) is defined to be \(E[g(X)]=\) \(\int_{A}^{B} g(x) f(x) d x .\) If the PDF of \(X\) is \(f(x)=\frac{15}{512} x^{2}(4-x)^{2}\), \(0 \leq x \leq 4\), find \(E(X)\) and \(E\left(X^{2}\right)\).

Find the area of the region in the first quadrant below \(y=e^{-x}\) above \(y=\frac{1}{2}\)

A continuous random variable \(X\) has PDF \(f(x)=\) \(\frac{3}{256} x(8-x), 0 \leq x \leq 8 .\) Find \(E\left(X^{2}\right)\) and \(E\left(X^{3}\right)\).

The circle \(x=a \cos t, y=a \sin t, 0 \leq t \leq 2 \pi\), is re- volved about the line \(x=b, 0

Using the same axes, draw the graphs of \(y=x^{n}\) on \([0,1]\) for \(n=1,2,4,10\), and \(100 .\) Find the length of each of these curves. Guess at the length when \(n=10,000\).

Use the Parabolic Rule with \(n=8\) to approximate the area of the region trapped between \(y=1-e^{-x^{2}}\) and \(y=e^{-x^{2}} .\)

Use the Parabolic Rule with \(n=8\) to approximate the area of the region trapped between \(y=\ln (x+1)\) and \(y=x / 4\). Hint: One point of intersection is obvious; the other you must approximate.

Find the area of the region trapped between \(y=\sin x\) and \(y=\frac{1}{2}, 0 \leq x \leq 17 \pi / 6\).