You want to find the volume of the solid obtained by rotating about the x-axis the region under the curve $\displaystyle {y}={x}^{{{1}/{4}}}$ from 0 to 5.

You first slice through the rotated solid at a generic point $\displaystyle {x}$ and get a circular cross-section. What is the area of the circular cross-section?

$\displaystyle {A}{\left({x}\right)}=$Preview Question 6 Part 1 of 3  

This makes the volume of the approximating disk with thickness $\displaystyle \Delta{x}$ equal to which expression?

Volume of disk = $\displaystyle \Delta{x}$ Preview Question 6 Part 2 of 3  

Now let $\displaystyle \Delta{x}$ approach 0, and sum the volumes of the infinitely many disks that approximate the solid of revolution. What total volume do you get?

$\displaystyle {V}={\int_{{0}}^{{5}}}{A}{\left({x}\right)}{\left.{d}{x}\right.}=$ Preview Question 6 Part 3 of 3