Describe Hooke’s Law in your own words.

Let $L$ be the lamina of uniform density $\rho=1$ obtained by removing circle $A$ of radius $r$ from circle $B$ of radius $2 r$ (see figure).

Find $M_y$ for $B$ and $M_y$ for $A$. Then use the previous part to compute $M_y$ for $L$.

Let $L$ be the lamina of uniform density $\rho=1$ obtained by removing circle $A$ of radius $r$ from circle $B$ of radius $2 r$ (see figure).

Show that $M_y$ for $L$ is equal to $\left(M_y\right.$ for $\left.B\right)-\left(M_y\right.$ for $\left.A\right)$.

Let $L$ be the lamina of uniform density $\rho=1$ obtained by removing circle $A$ of radius $r$ from circle $B$ of radius $2 r$ (see figure).

Show that $M_x=0$ for $L$.

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

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

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

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 the entire chain.

find the area of the region by integrating (a) with respect to and (b) with respect to (c) Compare your results. Which method is simpler? In general, will this method always be simpler than the other one? Why or why not? x=4-y² , x=y-2

A circular plate of radius r feet is submerged vertically in a tank of fluid that weighs w pounds per cubic foot. The center of the circle is k feet below the surface of the fluid, where k > r. Show that the fluid force on the surface of the plate is F = wk(πr²). (Evaluate one integral by a geometric formula and the other by observing that the integrand is an odd function.)

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

Use a graphing utility to graph the region bounded by the graphs of the equations. Use the integration capabilities of the graphing utility to approximate the centroid of the region. y = 5∛400-x², y = 0

(a) sketch the graph of the function, highlighting the part indicated by the given interval, (b) find 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, and (c) use the integration capabilities of a graphing utility to approximate the arc length. x = e^(-y), 0 ≤ y ≤ 2

Write and evaluate the definite integral that represents the area of the surface generated by revolving the curve on the indicated interval about the x-axis. $y=\frac{x^{3}}{18}$, $3 \leq x \leq 6$