Math Help

Example of Volume of Revolution using Method of Disks

Use the method of disks to calculate the volume generated by rotating the curves y = x

and y = √ x about the line y = 1. Recall the three step process.

Step 1: Identify the element of volume, dV, and show it on the graph of y = f(x)

Step 2: Determine the limits of integration for the region (volume to be calculated)

Step 3: Evaluate the integral

Step 1 The element of volume, dV = (πr₂² – πr₁²) dx.

Note that the “donut” is generated by rotating the hatched area about the axis of rotation.

Here r₁ = 1 ˗ y on the curve y = √x and r₂ = 1 ˗ y on the line y = x.

So r₁ = 1 ˗ √ x and r₂ = 1 ˗ x

Step 2 Limits of integration are from x = 0 to x = 1.

Step 3

x=1

V = ∫ (π [( 1 ˗ √x)² ) – ( 1 – x)²] dx

x=0

x=1

V = π ∫ ([( x + 1 - 2√x - x² + 2x - 1] dx = π / 6 (result)

x=0