How to find volume of the given solid analytically?

Question

Here is the question -

I am able to visualize the solid, but how do I find its volume? I'm unable to figure out the 2D structure that when rotated, produces this solid. Please help.

Edit: The answer required is 8/3

Answer 1

This solid is not obtained by a rotation.

Here, the area of the triangle $T_y$ that cross the y axis at $y$ is given by $A(y) = \frac{1}{2}(2x)^2 = 2(1-y^2) $, because $x^2+y^2=1$

Hence the volume of your solid is

$$\int_{-1}^1 A(y) dy = \int_{-1}^1 2(1-y^2)dy = \frac{8}{3}$$

Answer 2

Hint: the area of each of these triangles is half the height of a circle squared at the given point.

Hint #2: The height of the circle is going to be $2\sqrt{1-x^2}$, right? But we can figure out the area of the triangle only from the base!