Ximpledu

Volume of a Solid of Revolution

See how to find the volume of a solid of revolution
(disc method, washer method).
5 examples and their solutions.

Solid of Revolution

Definition

A solid of revolution is a 3D figure
formed by rotating a 2D figure (line, curve ...)
around an axis.

Disc Method

Formula


V = abπy2 dx
V = abS(x) dx
= abπy2 dx
Cross sectional area S(x): πy2
The shape of πy2 is a disc. (= circle)
→ Disc method

Volume from its Cross Sectional Area
Area of a Circle

V = abπx2 dy
Cross sectional area S(y): πx2

Example

A region is bounded by
y = x4, x = 1, and the x-axis.
If the region is rotated around the x-axis,
find the volume of the solid of revolution.
Solution

Example

A region is bounded by
y = x4, y = 1, and the y-axis.
If the region is rotated around the y-axis,
find the volume of the solid of revolution.
Solution

Example

Prove the volume of a sphere formula.
V = 43πr3
(r: Radius of the sphere)
Solution

Washer Method

Formula

To find the solid of revolution that has an empty space inside,
find the volume of the outer solid of revolution,
then subtract the volume of the inner solid of revolution.

V = abπ(y1)2 dx - abπ(y2)2 dx
Cross sectional area: π(y1)2 - π(y2)2
The shape is a washer: ⭗
→ Washer method

Example

A region is bounded by
y = ex, the tangent of y = ex at (1, e), and the y-axis.
If the region is rotated around the x-axis,
find the volume of the solid of revolution.
Solution

Example

A region is bounded by
y = 2√x - 1, the tangent of the function at (2, 2), and the x-axis.
If the region is rotated around the x-axis,
find the volume of the solid of revolution.
Solution