Transcription
We know that the volume of a sphere is 4/3 * its radius cubed. But where does this come from? Let's prove it.
In general, the volume of a solid is its cross-sectional area times its length. So, for example, if we have a cylinder, the cross-section is a circle. So, the volume is pi * r^2 * h. If we have a rectangular box, then the cross-section is a rectangle. The volume is length times width times height.
But the problem with this sphere is that its cross-section is not uniform. You will always get a circular cross-section. But the size of the circle varies depending on where you slice the sphere. So how do we solve this problem?
Let's take the sphere to be centered at the origin with radius r. Let's slice the sphere with a plane perpendicular to the x-axis at a certain point. The result is a circular cross-section whose radius depends on this position x. We can express this radius y in terms of x using the Pythagorean theorem.
And now we'll find the volume of the sphere by adding up all of the differential volumes of the slices. Think of dV as A(x) dx where A(x) is the cross-sectional area of this slice, which is pi * y^2, and dx is a differential thickness along the x-axis. So what we're doing is adding all of the slice volumes, eventually giving us the full volume of the sphere.
So we'll integrate pi * y^2 dx. The bounds of integration are from -r to r, and y^2 is r^2 - x^2. This is an even integrand. So we can integrate it from 0 to r and multiply by 2. The lower bound is zero. So all we have to do is plug in r into the anti-derivative, giving us r^3 - r^3 / 3, which is 2 * r^3 / 3, which gives us 4/3 * pi * r^3. So there you have it.