# Volume by revolution

• Feb 2nd 2008, 01:13 AM
Cursed
Volume by revolution
Let S be the region in the first quadrant bounded by the graphs of $\displaystyle y = x^1$$\displaystyle ^/$$\displaystyle ^5$ and $\displaystyle y = x^2$ Region S is the base of a solid whose cross sections perpendicular to the x-axis are squares. Find the volume of the solid.
• Feb 2nd 2008, 04:11 AM
curvature
Quote:

Originally Posted by Cursed
Let S be the region in the first quadrant bounded by the graphs of $\displaystyle y = x^1$$\displaystyle ^/$$\displaystyle ^5$ and $\displaystyle y = x^2$ Region S is the base of a solid whose cross sections perpendicular to the x-axis are squares. Find the volume of the solid.

Which is the axis of revolution?
• Feb 2nd 2008, 04:16 AM
galactus
This is not a revolution problem. The cross sections are perp. to the x-axis, so it looks like we integrate wrt x by using the area of the squares.

Since the cross-sections are perp. to the x-axis, then the squares will have base width y and area y^2.

But $\displaystyle y=x^{\frac{1}{5}}-x^{2}$

So, we need $\displaystyle \int_{0}^{1}[x^{\frac{1}{5}}-x^{2}]^{2}dx$