Use the disk method or the shell method to find the volumes of the solids generated by revolving the region bounded by the graphs of the equations about the given lines.

$$y=x^3, \quad y=0, \quad x=2$$

x=4

A circular porthole that has a diameter of 1 foot. Find the fluid force on the porthole, assuming that the center of the square is 15 feet below the surface.

Use the disk method or the shell method to find the volumes of the solids generated by revolving the region bounded by the graphs of the equations about the given lines.

$$y=x^3, \quad y=0, \quad x=2$$

y-axis

Approximate the arc length of the graph of the function over the interval [0, 4] in four ways. (a) Use the Distance Formula to find the distance between the endpoints of the arc. (b) Use the Distance Formula to find the lengths of the four line segments connecting the points on the arc when x=0, x=1, x=2, x=3, and x=4, Find the sum of the four lengths. (c) Use Simpson’s Rule with n=10 to approximate the integral yielding the indicated arc length. (d) Use the integration capabilities of a graphing utility to approximate the integral yielding the indicated arc length. f(x) = x³

y=x³, y=0, x=2 (a) the x-axis (b) the y-axis (c) the line x=4

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis. $y=e^{-3x}$, y=0, x=0, x=2

Find the area of the region by integrating with respect to $y$.

$$x=4-y^2$$

$$y=x^2$$

consider a 20-foot chain that weighs 3 pounds per foot hanging from a winch 20 feet above ground level. Find the work done by the winch in winding up the specified amount of chain. -Wind up one-third of the chain.

Find the area of the region by integrating with respect to $x$

$$x=4-y^2$$

$$x=y-2$$

A gasoline tank is an oblate spheroid generated by revolving the region bounded by the graph of $\frac{x^{2}}{16}+\frac{y^{2}}{9}=1$ about the y-axis, where x and y are measured in feet. How much gasoline can the tank hold?

Write a definite integral that represents the arc length of the curve over the indicated interval and observe that the integral cannot be evaluated with the techniques studied so far.

$x=e^{-y}, \quad 0 \leq y \leq 2$

Sketch the graph of the function, highlighting the part indicated by the given interval.

$x=e^{-y}, \quad 0 \leq y \leq 2$

In Exercise given below, find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the given axis.

$y=x^2, y=x^5, x$-axis

Find the area of the given region. f(y) = y(2-y), g(y) = -y