Volumes of Solids of Revolution
Calculate volumes formed when a region is rotated about the x-axis or y-axis using the disk method, a key HSC topic.
Rotation About the x-axis
When a region between y = f(x) and the x-axis is rotated 360 degrees about the x-axis, it creates a solid of revolution. Each cross-section perpendicular to the x-axis is a disk (circle) with radius y = f(x).
V = π ∫ab [f(x)]2 dx = π ∫ab y2 dx
The volume is π times the integral of the square of the function.
Why y2?
Each thin disk has radius r = y and thickness dx. The volume of one disk is πr2 · dx = πy2 dx. Summing all these disks from a to b gives the integral above.
Rotation About the y-axis
When the region between x = g(y) and the y-axis is rotated about the y-axis, the formula becomes:
V = π ∫cd [g(y)]2 dy = π ∫cd x2 dy
Express x as a function of y, and integrate with respect to y.
The limits c and d are now y-values, not x-values. You may need to rearrange the equation to express x in terms of y.
Example: Rotate y = √x about the y-axis from y = 0 to y = 2
Step 1: Rearrange: y = √x means x = y2.
Step 2: V = π ∫02 (y2)2 dy = π ∫02 y4 dy
Step 3: = π [y5/5]02 = π(32/5)
Answer: V = 32π/5 units3
Setting Up the Integral
The key steps for any volume of revolution problem are:
Sketch the region and identify the axis of rotation.
Choose the correct formula: π∫ y2 dx for x-axis rotation, or π∫ x2 dy for y-axis rotation.
Square the function and integrate between the correct limits.
Multiply by π at the end (do not forget this step!).
Key Vocabulary
Solid of Revolution
A 3D shape formed by rotating a 2D region around an axis.
Disk Method
Calculating volume by summing the volumes of thin circular disks (πr2 dx).
Axis of Rotation
The line (x-axis or y-axis) about which the region is rotated.
Cross-section
A slice of the solid perpendicular to the axis of rotation, forming a circle.
Worked Examples
Find the volume when y = x2 is rotated about the x-axis from x = 0 to x = 2.
Step 1: V = π ∫02 (x2)2 dx = π ∫02 x4 dx
Step 2: = π [x5/5]02 = π(32/5 − 0)
Answer: V = 32π/5 units3
Find the volume when y = √x is rotated about the x-axis from x = 0 to x = 4.
Step 1: y2 = x (since y = √x)
Step 2: V = π ∫04 x dx = π [x2/2]04 = π(8 − 0)
Answer: V = 8π units3
Find the volume when x = 3y is rotated about the y-axis from y = 0 to y = 2.
Step 1: x2 = 9y2
Step 2: V = π ∫02 9y2 dy = 9π [y3/3]02 = 9π(8/3)
Answer: V = 24π units3
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
The formula for volume when rotating y = f(x) about the x-axis is:
Question 2
Find the volume when y = 3 (a horizontal line) is rotated about the x-axis from x = 0 to x = 5.
Question 3
Find the volume when y = x is rotated about the x-axis from x = 0 to x = 3.
Question 4
When rotating about the y-axis, you must express:
Question 5
Find the volume when y = ex is rotated about the x-axis from x = 0 to x = 1.
Key Concepts Summary
- ● Rotation about x-axis: V = π ∫ y2 dx.
- ● Rotation about y-axis: V = π ∫ x2 dy (express x in terms of y).
- ● Each cross-section is a disk (circle) with area πr2.
- ● Remember to square the function before integrating.
- ● The π factor sits outside the integral — do not forget it.