Calculate the work required to stretch the following springs 1.25 m from their equilibrium positions. Assume Hooke's law is obeyed. A spring that requires a force of $250 \mathrm{~N}$ to be stretched $0.5 \mathrm{~m}$ from its equilibrium position.

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

$$y=x, y=2 \sqrt{x}$$

If necessary, use technology to evaluate or approximate the integral.

y=$\frac { 1 } { x }$ on [1, 10]

Write and simplify the integral that gives the arc length of the following curves on the given interval.

y=$\frac { 1 } { x }$ on [1, 10]

A data collection probe is dropped from a stationary balloon, and it falls with a velocity (in m/s) given by $v(t)=9.8t$, neglecting air resistance. After 10 s, a chute deploys and the probe immediately slows to a constant speed of 10 m/s, which it maintains until it enters the ocean.

Graph the velocity function.

Sketch the following regions (if a figure is not given) and find the total area by integrating with respect to y.

The region bounded by $x=y^2-3 y+12$ and $x=-2 y^2-6 y+30$

Evaluate the following limits Use l'Hopital 's Rule when convenient.

$\lim _{x \rightarrow 0}\left(\frac{1}{x}-\frac{\sin x}{x^2}\right)$

Calculate the work required to stretch the following springs 1.25 m from their equilibrium positions. Assume Hooke's law is obeyed. a. A spring that requires 100 J of work to be stretched 0.5 m from its equilibrium position. b. A spring that requires a force of 250 N to be stretched 0.5 from its equilibrium position.

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.

$y=\sqrt{\cos ^{-1} x}$, in the first quadrant

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. 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 following curves. Use the shell method to find the volume of the solid generated when R is revolved about indicated axis. $y=x^{3}-x^{8}+1$ and y=1; about the y-axis

Determine the area of the shaded region in the following figures by integrating with respect to y.

$$x=y^2-3 y+12$$

$$x=-2 y^2-6 y+30$$

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. y=

$$\sqrt { 50 - 2 x ^ { 2 } }$$

, in the first quadrant