AP Calculus AB · Unit 8 Applications of Integration
Learning Objectives
The Disk Method: Revolve a 2D shape with NO gaps from the axis of revolution. Sum up solid cylinders $V = \pi \int [R(x)]^2 dx$.
The Washer Method: Revolve a 2D shape with a GAP. Subtract the inner empty disk from the outer disk $V = \pi \int ([R_{out}]^2 - [r_{in}]^2) dx$.
Axis Shifts: Revolving around $y=-1$ instead of $y=0$ forces you to manually calculate $R(x)$ and $r(x)$ by subtracting the functions from the axis of revolution.
Tags
IntegrationVolumeDisk MethodWasher Method
OUTER / INNER BOUNDARIES $Top: y = \sqrt{x}$ $Bot: y = x^2$
Select Axis of Revolution
Interactive Solid Sweeper
Sweep Cross Section $x$ = 0.50
Volumetric Formula Breakdown
$V = \pi \int_{0}^{1} ([R(x)]^2 - [r(x)]^2) dx$
RADIANS EVALUATED AT X=0.50
$R_{out} =$0.71
$r_{in} =$0.25
Washer Area = $\pi(R^2 - r^2)$0.00
TOTAL INTEGRATED VOLUME
Total Volume $V =$0.00
Washer Method: Because there is a gap between $y=x^2$ and the $x$-axis, the revolution creates a solid with a hollow hole inside it. The volume requires subtracting the inner void cylinder from the outer cylinder!