IBHardCalculus10–12

Volume Swept by Rotating a Square Root Curve

Volume Gerado pela Rotação de uma Curva Raiz Quadrada

The region between the curve $y = \sqrt{x}$ , the $x$ -axis and the lines $x = 0$ and $x = 4$ is rotated through $360^\circ$ about the $x$ -axis. Find the exact volume of the solid formed.

Slice the solid perpendicular to the axis of rotation: each slice is a disc whose radius is the height of the curve. Sum the disc areas with an integral, and square the function before integrating, not after.

Solution

Step 1 of 6

1.Each disc has radius $y = \sqrt{x}$ and area $\pi y^{2} = \pi x$ :