Question

A region is bounded by y = square root of x, the x-axis, and x = 4. Write the integral that represents the volume of this region revolved about the line y = 3

Answer 1

$V=\int_0^4{\pi \cdot 3^2dx}-\int_0^4{\pi \cdot \left( 3-\sqrt{x} \right) ^2dx}=\\=\pi \int_0^4{\left( 6\sqrt{x}-x \right) dx}=\pi \left( 6\cdot \frac{2}{3}\cdot 4^{3/2}-\frac{4^2}{2} \right) =24\pi$