Volume of a sphere Let $R$ be the region bounded by the upper half of the circle $x^{2}+y^{2}=r^{2}$ and the $x$ -axis. A sphere of radius $r$ is obtained by revolving $R$ about the $x$ -axis.
a. Use the shell method to verify that the volume of a sphere of radius $r$ is $\frac{4}{3} \pi r^{3}$ . b. Repeat part (a) using the disk method.

Let R be the region bounded by the following curves. Let S be the solid generated when R is revolved about the given axis. If possible, find the volume of S by both the disk/washer and shell methods. Check that your results agree and state which method is easier to apply. y=x,

$$y=x^{1/3}$$

in the first quadrant; revolved about the x-axis

Let R be the region bounded by the following curves. Let S be the solid generated when R is revolved about the given axis. If possible, find the volume of S by both the disk/washer and shell methods. Check that your results agree and state which method is easier to apply. y=

$$\sqrt { \ln x } , y = \sqrt { \ln x ^ { 2 } }$$

, and y=1; revolved about the x-axis

(II) A thin $7.0\text{-kg}$ wheel of radius $34 \text{~cm}$ is a uniform disk and is weighted to one side by a $1.50\text{-kg}$ weight, small in size, placed $24 \text{~cm}$ from the center of the wheel. Calculate $(b)$ the moment of inertia about an axis through its $\text{\scriptsize{CM}}$, perpendicular to its face.

Let R be the region bounded by the following curves. Let S be the solid generated when R is revolved about the given axis. If possible, find the volume of S by both the disk/washer and shell methods. Check that your results agree and state which method is easier to apply. y=

$$x^2$$

, y=2-x, and x=0 in the first quadrant; revolved about the y-axis.

Find the potential Φ(r, θ) in the unit disk r<1 having the given boundary values Φ(1, θ). Using the sum of the first few terms of the series, compute some values of Φ and sketch a figure of the equipotential lines. Φ(1, θ)=1 if -½π<θ<½π and 0 otherwise

A disk drive manufacturer sells storage devices with capacities of one terabyte, 500 gigabytes, and 100 gigabytes with probabilities 0.5, 0.3, and 0.2, respectively. The revenues associated with the sales in that year are estimated to be $50 million,$25 million, and $10 million, respectively. Let X denote the revenue of storage devices during that year. Determine the probability mass function of X.

For each of the transactions given below, add the read- and write-actions to the computation and show the effect of the steps on main memory and disk. Assume that initially $A=5$ and $B=10$. Also, tell whether it is possible, with the appropriate order of OUTPUT actions, to assure that consistency is preserved even if there is a crash while the transaction is executing.
a) $A:=A+B ; B:=A+B$;
b) $B:=A+B ; A:=A+B$;
c) $A:=B+1 ; B:=A+1$;

a. Find the vertex and intercepts.
b. Sketch the graph.
c. Confirm your sketch using PLOT CONIC from the accompanying disk, or another suitable graphing program.
Find the vertex and intercepts of the following parabolas, and sketch the graph. Find decimal approximations for any radicals you encounter.
$y=\frac{1}{5} x^2+2 x-\frac{11}{5}$

A cylindrical metal rod is 1.60 m long and 5.50 mm in diameter. The resistance between its two ends (at 20$^\circ$C) is $1.09 \times 10^{-3} \Omega$. A round disk, 2.00 cm in diameter and 1.00 mm thick, is formed of the same material. What is the resistance between the round faces, assuming that each face is an equipotential surface?

Describe the range of each of the following functions. a) $f ( z ) = z + 5 \text { for } \operatorname { Re } z > 0$ b) $g ( z ) = z ^ { 2 }$ for z in the first quadrant, $\operatorname { Re } z \geq 0 , \operatorname { Im } z \geq 0$ c) $h ( z ) = \frac { 1 } { z } \text { for } 0 < | z | \leq 1$ d) $p ( z ) = - 2 z ^ { 3 }$ for z in the quarter-disk $| z | < 1,0 < \operatorname { Arg } z < \frac { \pi } { 2 }$

In a soapbox derby race, the cars have no engines; they simply coast down a hill to race with one another. Suppose you are designing a car for a coasting race. Do you want to use large wheels or small wheels? Do you want to use solid disk-like wheels or hoop-like wheels? Should the wheels be heavy or light?

You have a grindstone (a disk) that is 90.0 kg, has a 0.340-m radius, and is turning at 90.0 rpm, and you press a steel axe against it with a radial force of 20.0 N. (a) Assuming the kinetic coefficient of friction between steel and stone is 0.20, calculate the angular acceleration of the grindstone. (b) How many turns will the stone make before coming to rest?

Assume C is a circle centered at the origin, oriented counterclockwise, that encloses disk R in the plane. Complete the following steps for each vector field F. a. Calculate the two-dimensional curl of F. b. Calculate the two-dimensional divergence of F. c. ls F irrotational on R? d. Is F source free on R? $\mathbf{F}=\left\langle x^{2}+2 x y,-2 x y-y^{2}\right\rangle$