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.

Let R be the region bounded by the following curves. Use the disk method to find the volume of the solid generated when R is revolved about the x-axis.

$$y = \frac { 1 } { \sqrt { 1 + x ^ { 2 } } }$$

, y=0, x=-1, and x=1

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=

$$( x - 2 ) ^ { 3 }$$

-2, x=0, and y=25; revolved about the y-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=6/(x+3), y=2-xl revolved about the x-axis

Let R be the region bounded by the following curves. Use the disk method to find the volume of the solid generated when R is revolved about the x-axis.

$$y = \frac { 1 } { \sqrt [ 4 ] { 1 - x ^ { 2 } } } , y = 0 , x = - \frac { 1 } { 2 }$$

, and x = $\frac { 1 } { 2 }$

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-

$$x^4$$

, y=0; revolved about the x-axis

In this exercise, evaluate the following integral. A sketch is helpful.

$\displaystyle\iint_R x y d A ; R$ is bounded by $x=0, y=2 x+1$, and $y=-2 x+5$.

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=1/(x+1), y=1-x/3; revolved about the x-axis

Let R be the region bounded by the following curves. Use the disk method to find the volume of the solid generate when R is revolved about the x-axis. y=sin x on

$$[ 0 , \pi ]$$

, y=0 (Recall that

$$sin^2$$

x=

$$\frac { 1 } { 2 }$$

(1-cos 2x).)

Disk method Let R be the region bounded by the following curves. Use the disk method to find the volume of the solid generated when R is revolved about the $x$-axis.
$y=4-x^2, y=0, x=0$

$$∫∫R y^2dA$$

, where R is the region bounded by the curves

$$xy = 1, xy=2, xy^2=1, xy^2=2; u=xy, v=xy^2$$

. Illustrate by using a graphing calculator or computer to draw R

Disks/washers about the $\boldsymbol{y}$-axis Let $R$ be the region bounded by the following lines and curves. Use the disk or washer method to find the volume of the solid generated when $R$ is revolved about the y-axis. \

$$y=0, y=\sqrt{x-1}, y=2, x=0$$

Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given line. $x=0, y=\sqrt{x}$, and y=2; about x=4

Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis. $y=x^{2}, y=2-x$, and y=0; about the y-axis