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

$$y = \sqrt { x }$$

, y=0, 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=sin x on $[0, \pi]$ and y=0; about the x axis (Hint. Recall that $\sin ^{2} x=\frac{1-\cos 2 x}{2}$.)

Let R be the region bounded by the curves y = 1 - x^2 and y = x^6 - x + 1. Estimate the following quantities. The area of R.

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

Calculate the flux of the vector field through the surface.

$\vec{F} = \cos(x^2 + y^2)\vec{k}$ through the disk $x^2 + y^2 \leq 9$ oriented upward in the plane $z = 1$.

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

$$y=x^2$$

, x=1, and y=0. Use the shell method to find the volume of the solid generated when R is revolved about the following lines. y=2

Use the shell method to set up and evaluate the integral that gives the volume of the solid generated b revolving the plane region about the y-axis. \

$$y=2 x, \quad y=4, \quad x=0$$

Use the shell method to find the volumes of the solids generated by revolving the regions bounded by the given curves about the given lines.

$$y=x^{4}, \quad y=4-3 x^{2}$$

The x-axis

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

$$y^2$$

, x=0, and y=3

Decide whether it is more convenient to use the disk method or the shell method to find the volume of the solid of revolution. Explain your reasoning. (Do not find the volume.) (y-2)² = 4-x

Use the shell method to find the volumes of the solids generated by revolving the regions bounded by the given curves about the given lines.

$$y=x^{3}, \quad y=8, \quad x=0$$

The x-axis

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

$$y=x^3$$

, y=0, x=2

Let $R$ be the region bounded by the following curves. Use the disk or washer method to find the volume of the solid generated when $R$ is revolved about the y-axis. $y=x, y=2 x, y=6$