Disc Integration

Disc Integration

How to find the volume of a rotated region by using the disc integration: formulas (x-axis, y-axis), examples, and their solutions.

Formula 1: Rotated Around the x-axis

V = [inteval, from a to b, of (pi*y^2) dx]

If the region under a graph is rotated
around the x-axis,
then the cross-sectional area (= sliced area) is a circle,
whose area is πy2 (= π[f(x)]2).

So the volume of the 3D figure is
V = ∫ab πy2 dx.

Finding volume from its slices

The sliced area is a disk.
So this integration is called the disc integration.

Example 1

A region is bounded by y = x^4, x = 1, and the x-axis. If the region is rotated around the x-axis, find the volume of the rotated region.

First find the bounded region.

This gray colored region is the bounded region.

Draw the rotated figure
that is rotated around the x-axis.

And draw the sliced disk at x = x.

The disk's radius is y, which is x4.

So V = ∫01 π(x4)2 dx.

Take π out from the integral.

And solve the integral.

Definite integration of polynomials

Formula 2: Rotated Around the y-axis

V = [inteval, from a to b, of (pi*x^2) dy]

If the region under a graph is rotated
around the y-axis,
then the cross-sectional area (= sliced area) is a circle,
whose area is πx2 (= π[g(y)]2).

So the volume of the 3D figure is
V = ∫ab πx2 dy.

Example 2

A region is bounded by y = x^4, y = 1, and the y-axis. If the region is rotated around the y-axis, find the volume of the rotated region.

First find the bounded region.

This gray colored region is the bounded region.

And, for the integration,
change y = x4 into x = y1/4.

Draw the rotated figure
that is rotated around the y-axis.

And draw the sliced disk at y = y.

The disk's radius is x, which is y1/4.

So V = ∫01 π(y1/4)2 dy.

Take π out from the integral.

And solve the integral.

Definite integration of polynomials

Example 3: Proof of the Formula (Volume of a Sphere)

If the radius of a sphere is r, show that the voulme of the sphere is (4/3)*pi*r^3.

Start from the equation of a circle
whose radius is r:
x2 + y2 = r2.

Equation of a circle

This circle will be rotated.

Draw the rotated circle
that is rotated around the x-axis.

Draw the sliced disk at x = x.
The disk's radius is y.

And you have y2 = r2 - x2.

So V = ∫-rr πy2 dx
= ∫-rr π(r2 - x2) dx.

π can get out from the integral.

The integral inteval is symmetric: [-r, r].
And r2 (= constant) and -x2 are all even functions.

So write 2π in front of the integral
and change the integral limits into ∫0r.

Definite integration of even functions

Solve the integral (with respect to x).

Definite integration of polynomials

Then V = (4/3)πr3.

This is the proof of the volume of a sphere formula.