Question

(1 point) The volume of the solid obtained by rotating the region enclosed by x=2y,y3=x(with y≥0) about the y-axis can be computed using the method of disks or washers via an integral V=∫ba ? with limits of integration a= and b= . The volume is V= cubic units. Note: You can earn full credit if the last question is correct and all other questions are either blank or correct.

Answer 1

`x=2y` and
`x=y^3` intersect when

`2y=y^3\implies y^3-2y=y(y^2-2)=y(y-\sqrt2)(y+\sqrt2)=0`

`\implies y=0\text{ or }y=\sqrt2\text{ or }y=-\sqrt2`

We omit the negative root, since we only care for
`y\ge0`.

In the interval (0, √2), we have
`y^3<2y`. Then the volume is given by the integral

`\displaystyle\pi\int_0^(\sqrt2)(2y)^2-(y^3)^2\,\mathrm dy=\pi\int_0^(\sqrt2)4y^2-y^6\,\mathrm dy`

and so the volume is (32√2)/21 π.