Unit 3B Practice Exam B

Find the area enclosed by $y=x+2$ and $y=x^2$.

Rotate the region under $y=1-x^2$, $0\le x\le1$, around the $x$-axis. Find the volume.

Rotate the same region around the $y$-axis using cylindrical shells.

A solid has semicircular cross-sections perpendicular to the $x$-axis. The diameter at $x$ is $2\sqrt{x}$ for $0\le x\le4$. Find the volume.

Find the surface area generated by rotating $y=x$, $0\le x\le1$, around the $x$-axis.

A rod occupies $[0,1]$ and has density $\lambda(x)=1+x^2$. Find its mass and center of mass.

A spring has constant $k=200\,\mathrm{N/m}$. Find the work required to stretch it from 0.10 m to 0.30 m beyond equilibrium.

A vertical cylindrical tank has radius 1 m and height 2 m and is full of water. Pump the water to an outlet 1 m above the top. Set up and evaluate the work.

A triangular gate has its top edge at the water surface, width 4 m at the surface, and vertex 3 m below the surface. Find the hydrostatic force.

If $C'(q)=15+0.02q$ and $C(0)=2000$, find $C(300)$.

The density $p(x)=3x^2$ is defined on $[0,1]$. Verify normalization and find $E[X]$.

For the region under $y=1-x^2$ on $[0,1]$, explain why shells are natural about the $y$-axis while disks are natural about the $x$-axis.

Critique this statement: "Once an integral is written, units and geometry no longer matter." Give at least two concrete reasons it is false.