AcademyApplications Of Integrals
Academy
Volumes
Level 1 - Calculus topic page in Applications Of Integrals.
Principle
Volumes from calculus are found by slicing a solid into thin cross-sections and integrating cross-sectional area. The formula depends on the geometry of the slices.
Notation
\(V\)
volume
\(A(x)\)
cross-sectional area at x
\(R(x),r(x)\)
outer and inner radii for washers
The Core Method
Use the slice formula.
Volume by cross-sections
\[V=\int_a^b A(x)\,dx\]
For rotation about a horizontal axis with washers,
Washer method
\[V=\pi\int_a^b\left(R(x)^2-r(x)^2\right)\,dx\]
Worked Cases
Question
Find the volume formed by rotating \(y=x\) from \(x=0\) to \(x=2\) about the \(x\)-axis.
Answer
A slice perpendicular to the \(x\)-axis forms a disk with radius \(R(x)=x\). Thus \(A(x)=\pi x^2\). The volume is \(V=\pi\int_0^2 x^2dx=\pi[x^3/3]_0^2=8\pi/3\).
Examples
Question
Set up the volume from rotating the region between \(y=2\) and \(y=x\), \(0\le x\le1\), about the \(x\)-axis.
Answer
The outer radius is \(R=2\), and the inner radius is \(r=x\). The washer formula gives \(V=\pi\int_0^1(2^2-x^2)dx\).