Unit 3B Cumulative Practice

Find the area enclosed by $y=x^2$ and $y=2x+3$.

Rotate that same enclosed region about the horizontal line $y=-1$. Set up and evaluate the washer integral.

Rotate that same enclosed region about the vertical line $x=-2$. Set up and evaluate the shell integral.

The base is the region enclosed by $y=x^2$ and $y=2x+3$. Cross-sections perpendicular to the $x$-axis are squares. Find the volume.

Find the arc length of $y=x^{3/2}$ on $[0,4/9]$.

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 $0.10\,\mathrm{m}$ to $0.30\,\mathrm{m}$ beyond equilibrium.

A $4\,\mathrm{m}$-long triangular trough has ends $2\,\mathrm{m}$ deep and $3\,\mathrm{m}$ wide at the top. It is full of water. Set up and evaluate the work to pump the water to the top edge using $9800\,\mathrm{N/m^3}$.

A triangular vertical gate has a $4\,\mathrm{m}$ top edge at the water surface and a vertex $3\,\mathrm{m}$ below. Find the hydrostatic force using $9800\,\mathrm{N/m^3}$.

Marginal cost is $C'(q)=15+0.02q$ dollars per unit and fixed cost is $C(0)=2000$. Find $C(300)$.

A probability density has the form $p(x)=kx$ on $[0,2]$. Find $k$, verify normalization, and compute $E[X]$.