Question

The base of a solid is the region in the first quadrant between the graph of y=x2 and the x -axis for 0≤x≤1 . For the solid, each cross section perpendicular to the x -axis is a semicircle. What is the volume of the solid?

Answer 1

Answer:

The volume of the solid is π/40 cubic units.

Step-by-step explanation:

Please refer to the graph below.

Recall that the area of a semi-circle is given by:

$\displaystyle A=\frac{1}{2}\pi r^2$

The volume of the solid will be the integral from x = 0 to x = 1 of area A. Since the diameter is given by y, then the radius is y/2. Hence, the volume of the solid is:

$\displaystyle V=\int_0^1\frac{1}{2}\pi \left(\frac{y}{2}\right)^2\, dx$

Substitute:

$\displaystyle V=\frac{1}{2}\pi\int_0^1\left(\frac{x^2}{2}\right)^2\, dx$

Simplify:

$\displaystyle V=\frac{1}{2}\pi \int_0^1\frac{x^4}{4}\, dx$

Integrate:

$\displaystyle V=\frac{1}{2}\pi \left[\frac{x^5}{20}\Big|_0^1\right]$

Evaluate:

$\displaystyle V=\frac{\pi}{40}\left((1)^5-\left(0\right)^5\right)=\frac{\pi}{40}\text{ units}^3$

The volume of the solid is π/40 cubic units.