Find the Volume of the solid

Question

The base of a solid is a circle with radius 1. Each cross-section perpendicular to a given diameter is a square. Find the volume of the solid.

Answer 1

Hint: I assume this is your setup.

(Large version)

Answer 2

You tell us that the radius of the circle is 1, so we know that this circle can be represented as $x^2+y^2=1$. When we solve this for $y$, we get $y=\sqrt{1-x^2}$. This represents the distance from the center of the base of the circle to its edge. To get the distance from edge to edge, we need to double this: $2y=2\sqrt{1-x^2}$.

The formula for the area of a square is $A=s^2$. We want to find the area of the cylinder using the area of its square cross sections. So, we want $s=2\sqrt{1-x^2}$.

To find the area of the cylinder using cross sections requires integration, because length of the sides of the cross sections is always changing. We want to integrate from -1 to 1, because the radius of the circle is 1. Try $$\int_{-1}^{1} A dx = \int_{-1}^{1} s^2 dx = \int_{-1}^{1} s^2 dx = \int_{-1}^{1} 4(1-x^2)dx. $$