Find the volume formed by rotating the region enclosed by:
x=4y , y^3 = x, with y ≥ 0 about the y-axis.
thanks for any help
Graphing this, we can see it is symmetrical about the y-axis.
Solve each for y in terms of x:
$\displaystyle y=\frac{x}{4}, \;\ y=x^{\frac{1}{3}}$
Set each equal and solve for x to find the limits of integration.
Doing this, we see we get x=-8,0,8
Using shells:
$\displaystyle 2{\pi}\int_{0}^{8}x(\frac{x}{4}-x^{\frac{1}{3}})dx$
Integrating over -8 to 0 will yield the same result.
Using washers, we can get the same thing.
$\displaystyle {\pi}\int_{0}^{2}\left[(4y)^{2}-(y^{3})^{2}\right]dy$