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

Write each number in standard notation. 9.2 X 10^6

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. 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=

$$x^2$$

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

A 1125-ft³ open-top rectangular tank with a square base x ft on a side and y ft deep is to be built with its top flush with the ground to catch runoff water. The costs associated with the tank involve not only the material from which the tank is made but also an excavation charge proportional to the product xy. If the total cost is c = 5(x² + 4xy) + 10xy, what values of x and y will minimize it?

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

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.
$y=4-x^2, y=0, x=0$

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

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. 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 y-axis

Parker says that any two congruent acute triangles can be arranged to make a rectangle. Tamika says that only two congruent right triangles can be arranged to make a rectangle. Is either of them correct? Explain your reasoning.

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. \

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

A gear on a bicycle has the shape of a circle. One gear has a diameter of 4 inches, and a smaller one has a diameter of 2 inches Justin says that the circumference of the larger gear is 2 inches more than the circumference of the smaller gear. Do you agree? Explain your answer.