Fins the volume if the solidobtained by rotating the region bounded by y=x^(1/3) and y=x/4
Rotating it about what?. y-axis?. x-axis?. Some other axis?.
Rotated about x-axis using washers:
Find the limits of integration by setting them equal and solving for x. We get x=0 and 8.
$\displaystyle {\pi}\int_{0}^{8}(\frac{x^{2}}{16}-x^{\frac{2}{3}})dx$
Using shells rotated about x-axis:
$\displaystyle 2{\pi}\int_{0}^{2}y(y^{3}-4y)dy$
Here is a happy little animated diagram showing the region rotated about the x-axis. Enjoy.
And yes, that is what I gave you. Both methods rotating about the x-axis. Shells and washers.
When we use shells, the cross sections are parallel to the axis about which we are revolving. Since they are parallel to the x-axis, we can picture them being stacked up the y-axis. That is why we integate w.r.t y in that case. We solve the equations for x in terms of y and change the limits to the y limits, which are 0 to 2.