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

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

The regions $R_1, R_2$, and $R_3$ (see figure) are formed by the graphs of $y=2 \sqrt{x}, y=3-x$, and $x=3$.

Find the volume of the solid obtained by revolving region $R_2$ about the $x$-axis.

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.

If the probe was released from an altitude of 3 km, when does it enter the ocean?

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.

How far does the probe fall in the first 30 s after it is released?

Use technology to evaluate or approximate the integral. $y=4 x-x^2$, for $0 \leq x \leq 4$

Consider the following curves on the given intervals. Use a calculator or software to approximate the surface area. $y=\sqrt{\sin \pi x}$, for $0 \leq x \leq \frac{1}{2}$; about the $x$-axis

Calculate the work required to stretch the following springs 0.4 m from their equilibrium positions. Assume Hooke's law is obeyed. A spring that requires $2 \mathrm{~J}$ of work to be stretched $0.1 \mathrm{~m}$ from its equilibrium position.

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.

Consider the following curves on the given intervals. a. Write the integral that gives the area of the surface generated when the curve is revolved about the given axis. b. Use a calculator or software to approximate the surface area. $y=\sqrt{\sin \pi x}$, for $0 \leq x \leq \frac{1}{2}$; about the x-axis

a. Write and simplify the integral that gives the arc length of the following curves on the given interval. b. If necessary, use technology to evaluate or approximate the integral. $y=4 x-x^{2}$, for $0 \leq x \leq 4$

Calculate the work required to stretch the following springs 0.4 m from their equilibrium positions. Assume Hooke's law is obeyed. a. A spring that requires a force of 50 N to be stretched 0.1 m from its equilibrium position. b. A spring that requires 2 J of work to be stretched 0.1 m from its equilibrium position.

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. $y=\sqrt{25-x^{2}}$ and y=0; about the x-axis (Verify that your answer agrees with the volume formula for a sphere)

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

$\lim _{x \rightarrow \infty} x^3\left(\frac{1}{x}-\sin \frac{1}{x}\right)$

The regions $R_1, R_2$, and $R_3$ (see figure) are formed by the graphs of $y=2 \sqrt{x}, y=3-x$, and $x=3$.

Find the volume of the solid obtained by revolving region $R_2$ about the $y$-axis.