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Code Scholars Tutoring · AP Calculus AB

Disks and washers about ordinary and shifted axes

Unit 8 · Topics 8.9, 8.10, 8.11, 8.12. Original AP Calculus AB notes, worked example, and self-check.

Understand the idea

For rotation, radius is distance from the rotation axis. A disk has one radius; a washer removes an inner disk from an outer disk. Both radii must be measured in the same slicing direction. A shifted axis changes the radius expressions, not the general formula. Choose vertical slices for rotation about a horizontal axis and horizontal slices for rotation about a vertical axis when using disks or washers.

V=π∫ab(R(x)2−r(x)2) dxV=\pi\int_a^b\left(R(x)^2-r(x)^2\right)\,dx

Worked example

Rotate the region between y=x\displaystyle y=x and y=x2\displaystyle y=x^2, 0≤x≤1\displaystyle 0\le x\le1, about the x\displaystyle x-axis.

The outer radius is x\displaystyle x and inner radius is x2\displaystyle x^2. The washer area is π(x2−x4)\displaystyle \pi(x^2-x^4), so volume is π(1/3−1/5)=2π/15\displaystyle \pi(1/3-1/5)=2\pi/15.

Try it yourself

Rotate the region below y=x\displaystyle y=x and above y=0\displaystyle y=0 for 0≤x≤1\displaystyle 0\le x\le1 about y=2\displaystyle y=2. Set up the volume.

Show solution and reasoning

The outer radius is 2\displaystyle 2 and inner radius is 2−x\displaystyle 2-x. The volume is π∫01[4−(2−x)2] dx=5π/3\displaystyle \pi\int_0^1[4-(2-x)^2]\,dx=5\pi/3.

Unit 8 MCQ