# 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

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 *π**y*^{2} (= *π*[*f*(*x*)]^{2}).

So the volume of the 3D figure is*V* = ∫_{a}^{b} *π**y*^{2} *dx*.

Finding volume from its slices

The sliced area is a disk.

So this integration is called the disc integration.

## Example 1

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 *x*^{4}.

So *V* = ∫_{0}^{1} *π*(*x*^{4})^{2} *dx*.

Take *π* out from the integral.

And solve the integral.

Definite integration of polynomials

## Formula 2: Rotated Around the *y*-axis

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 *π**x*^{2} (= *π*[*g*(*y*)]^{2}).

So the volume of the 3D figure is*V* = ∫_{a}^{b} *π**x*^{2} *dy*.

## Example 2

First find the bounded region.

This gray colored region is the bounded region.

And, for the integration,

change *y* = *x*^{4} into *x* = *y*^{1/4}.

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 *y*^{1/4}.

So *V* = ∫_{0}^{1} *π*(*y*^{1/4})^{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)

Start from the equation of a circle

whose radius is *r*:*x*^{2} + *y*^{2} = *r*^{2}.

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 *y*^{2} = *r*^{2} - *x*^{2}.

So *V* = ∫_{-r}^{r} *π**y*^{2} *dx*

= ∫_{-r}^{r} *π*(*r*^{2} - *x*^{2}) *dx*.

*π* can get out from the integral.

The integral inteval is symmetric: [-*r*, *r*].

And *r*^{2} (= constant) and -*x*^{2} are all even functions.

So write 2*π* in front of the integral

and change the integral limits into ∫_{0}^{r}.

Definite integration of even functions

Solve the integral (with respect to *x*).

Definite integration of polynomials

Then *V* = (4/3)*π**r*^{3}.

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