# Washer Method

How to find the volume of the rotated region by using the Washer method: formula, example, and its solution.

## Formula

If the region between two functions is rotated

around the *x*-axis,

then the cross-sectional area is an annulus.

(= ring shape)

If the outer radius is *y*_{1} [= *f*(*x*)]

and the inner radius is *y*_{2} [= *g*(*x*)]

then *V* = [*V*_{1}, outer figure] - [*V*_{2}, inner figure]

= ∫_{a}^{b} *π**y*_{1}^{2} *dx* - ∫_{a}^{b} *π**y*_{2}^{2} *dx*.

Disc integration

## Example 1

To find the rotate region,

first find the tangent line of *y* = *e*^{x} at (1, *e*).

Set *f*(*x*) = *e*^{x}.

Then *f*'(*x*) = *e*^{x}.

Derivative of *e*^{x}

Then *f*'(1) = *e*.

The tangent line's slope is *f*'(1) = *e*.

And the tangent line passes through (1, *e*).

So the tangent line is *y* = *e*(*x* - 1) + *e*,

which is *y* = *e**x*.

Tangent line to a graph

The gray colored region is the region

that is rotated around the *x*-axis.*y* = *e*^{x} is the outer function.

Its integral interval is [0, 1].*y* = *ex* is the inner function.

Its integral interval is also [0, 1].

So = ∫_{0}^{1} *π*(*e*^{x})^{2} *dx* - ∫_{0}^{1} *π*(*ex*)^{2} *dx*.

Solve the integrals.

Indefinite integration of *e*^{x}

Definite integration of polynomials

## Example 2

To find the rotate region,

first find the tangent line of *y* = 2√*x* - 1 at (2, 2).

Set *f*(*x*) = 2(*x* - 1)^{1/2}.

Then *f*'(*x*) = 2⋅(1/2)(*x* - 1)^{-1/2}⋅1.

Chain rule in differentiation

Power rule in differentiation (Part 3)

Then *f*'(2) = 1.

The tangent line's slope is *f*'(2) = 1.

And the tangent line passes through (2, 2).

So the tangent line is *y* = 1(*x* - 2) + 2,

which is *y* = *x*.

Tangent line to a graph

The gray colored region is the region

that is rotated around the *x*-axis.*y* = *x* is the outer function.

Its integral interval is [0, 2].*y* = 2(*x* - 1)^{1/2} is the inner function.

Its integral interval is [1, 2],

which is **different**

from the outer function's integral interval.

So = ∫_{0}^{2} *π*(*x*)^{2} *dx* - ∫_{1}^{2} *π*[2(*x* - 1)^{1/2}]^{2} *dx*.

Solve the integrals.

Definite integration of polynomials

To make the denominators the same,

multiply 3/3 by 2.