Problems

01

Shells volume - offset graph, $y$-axis

Consider the region in the first quadrant bounded by the lines $x=0$, $x=2$, $y=0$, and the curve $y=\frac{1}{\sqrt{x^2+1}}$. Revolve this about the $y$-axis.

Find the volume of the resulting solid.

02

Shells volume - set up integrals, both axes

Consider the region in the first quadrant bounded by the lines $x=0$ and $y=2$, and the curve $y=4-x^2$.

Set up integrals to find the volumes of the solids obtained by revolving this region about (i) the $x$-axis, and (ii) the $y$-axis.

(No need to evaluate the integrals in this problem.)

03

Shells volume - shells v. washers

Consider the region in the $xy$-plane, in the first quadrant, bounded by the $y$-axis on the left, by $y=8-6x^2$ on the top, and $y=2x^2$ on the bottom.

A 3D solid is given by revolving this region around the $y$-axis.

(a) Find the volume of the solid using the method of shells.

(b) Attempt to find the volume of the solid using the method of washers/disks. Why is this harder? (TWO reasons!)