# Thread: Finding the volume of a solid using it's equation?

1. ## Finding the volume of a solid using it's equation?

A solid has, as its base, the circular region in the xy-plane bounded by the graph of x2 + y2 = 4. Find the volume of the solid if every cross section by a plane perpendicular to the x-axis is a quarter circle with one of its radii in the base.

I don't know how to go about this? Do I make it a definite integral? I'm not sure. What am I supposed to get here?

2. ## Re: Finding the volume of a solid using it's equation?

one of its radii in the base
I assume this means that the radius goes from one side of the circle to the other for the given value of x which the plane lies on.

If the plane lies perpendicular to the x axis then is crosses the x axis at only one point.
For a given point x where the plane crosses the x axis the "height" of the circle is $\sqrt{4-x^2}$ (if you draw a 2D diagram of the circle with the y axis in the upwards direction then it is more clear what I mean by height)
The radius of the quarter circle at this point will be $2\sqrt{4-x^2}$.
The area of the quarter circle will be $\frac{1}{4} \pi r^2= \frac{1}{4} \pi (2\sqrt{4-x^2})^2$.

The volume of a small slice of the quarter circle is dV. $dV=\frac{1}{4} \pi (2\sqrt{4-x^2})^2 dx$.

Integrate the volume between the values of x where a plane crosses the circle to get the volume of the solid