Unit 3B Review: Applications of Integration

Find the area between $y=2x$ and $y=x^2$ on $0\le x\le2$ using vertical slices.

Set up and evaluate the same area between $y=2x$ and $y=x^2$ using horizontal slices.

For the region between $y=2x$ and $y=x^2$ on $0\le x\le2$, find the volume about the $x$-axis with washers and the volume about the $y$-axis with shells. Compare how radius and height are defined.

The base is the region under $y=\sqrt{4-x^2}$ above the $x$-axis for $-2\le x\le2$. Cross-sections perpendicular to the $x$-axis are semicircles whose diameters lie in the base. Find the volume.

Find the arc length of $y=2x+1$ on $[0,3]$, then verify the result with the distance formula between the endpoints.

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

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

A full conical tank is $4\,\mathrm{m}$ high with top radius $2\,\mathrm{m}$. Water is pumped to an outlet $1\,\mathrm{m}$ above the top. Set up and evaluate the work using $9800\,\mathrm{N/m^3}$.

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

Marginal cost is $C'(q)=20+0.04q$ dollars per unit and $C(100)=5000$. Find $C(200)$ and identify the accumulated added cost.