Unit 3B Practice Exam A

Find the area between $y=2x$ and $y=x^2$ on $[0,2]$.

The region bounded by $x=y^2$ and $x=y+2$ is described most naturally with horizontal slices. Find its area.

Rotate the region under $y=\sqrt{x}$, $0\le x\le4$, around the $x$-axis. Find the volume.

Use cylindrical shells to rotate the region under $y=4-x^2$, $0\le x\le2$, around the $y$-axis.

A solid has square cross-sections perpendicular to the $x$-axis. The side length is $2-x$ for $0\le x\le2$. Find the volume.

Find the arc length of $y=2x+1$ on $[0,3]$.

A rod occupies $[0,3]$ and has density $\lambda(x)=2+x$ kg/m. Find its mass and center of mass.

A force $F(x)=10+3x$ newtons moves an object from $x=0$ to $x=4$ meters. Find the work.

A rectangular tank is 4 m long, 3 m wide, and 2 m deep, and is full of water. Water is pumped to an outlet 1 m above the top. Set up and evaluate the work using water weight density $9800\,\mathrm{N/m^3}$.

A vertical rectangular gate is 2 m wide and extends from the water surface to a depth of 3 m. Find the hydrostatic force using water weight density $9800\,\mathrm{N/m^3}$.

If marginal cost is $C'(q)=20+0.04q$, find the added cost of increasing production from 100 to 200 units.

The probability density is $p(x)=2x$ on $[0,1]$. Find $P(X\le1/2)$ and $E[X]$.

For one problem above, explain how the units of the slice contribution become the units of the final answer. Also give one independent reasonableness check.