Find the area bounded by the parabola $y = x^2$ and the line $y = x + 2$.

Find the area of the region bounded by $x = y^2 - 4$ and $x = y + 2$.

Find the volume obtained by rotating the region bounded by $y = \sqrt{x}$, the x-axis, and $x=4$ around the x-axis.

Find the volume of the region bounded by $y=x$ and $y=x^2$ rotated around the x-axis.

Find the volume of the solid generated by rotating the region bounded by $y = 2x - x^2$ and the x-axis around the y-axis.

Find the arc length of the curve $y = \frac{2}{3}x^{3/2}$ from $x=0$ to $x=1$.

Find the centroid of the region bounded by $y = \sqrt{x}$ and $y = x^2$.

A conical tank (point facing down) has a radius of 4 meters at the top and a height of 10 meters. It is filled with water (density $\rho = 1000$ kg/m$^3$) to a depth of 8 meters. Find the work done in pumping all the water out over the top rim of the tank. Let $g = 9.8$ m/s$^2$.

A vertical rectangular dam is 20 meters wide and 15 meters high. If water is at the top edge of the dam, find the total hydrostatic force exerted on the dam by the water. (Density of water $\rho = 1000$ kg/m$^3$, $g = 9.8$ m/s$^2$).

Use the Second Theorem of Pappus to find the volume of a torus generated by rotating a circle of radius $r = 2$ centered at $(3, 0)$ around the y-axis.