AP Calculus AB / Applications of Integration
Lesson
Volumes from slices
Disks, washers, and known cross sections are all “area of a slice, then integrate.”
Learning goals
- Set up a disk or washer integral for a solid of revolution.
- Set up a volume with a stated cross-section shape.
Explanation
Revolve a region around a line and you stack circular slices. A disk has volume πR² dx (or dy). A washer has π(R_outer² − R_inner²). The radii are distances from the slice to the axis, not automatically the y-value on the paper. Draw a sample slice perpendicular to the axis of rotation.
If a problem gives the shape of a cross section—equilateral triangle, square, semicircle—write that slice’s area in terms of the segment length across the base region, then integrate. The habit is the same: A(x) dx, added from one end of the solid to the other.
Key terms
- Disk method. A volume setup using circular slices of radius equal to the distance from the axis to the curve.
- Washer method. A volume setup that subtracts an inner disk from an outer disk.
Common mistakes
- Using the function as a radius when the axis is shifted.
- Forgetting to square the radii, or subtracting the radii before squaring.
Practice
Original Marlow Works items. Check the answer explanation after you try.
What is lim (x→3) of (x² − 9) / (x − 3)?
Original Marlow Works item — not a College Board question.
Take your time—this is practice, not a test.
Answer explanation
You revolve around the y-axis. Should a typical slice be horizontal or vertical if you want disks or washers with dy?
Answer. Horizontal, so the slice is perpendicular to the y-axis and the thickness is dy.
The integration variable follows the axis of rotation for disks and washers.
Related resources
External links with reuse status. Marlow Works is independent and does not copy restricted exam or textbook material.
Official / link only
AP Calculus AB course page
Official eight-unit framework and 2027 exam updates.
College Board · All rights reserved · accessed 2026-10-01
Open sourceReference only — do not copy
OpenStax College Algebra 2e
Algebra reference; current terms include noncommercial and AI-ingestion limits.
OpenStax · Restricted — see publisher page · accessed 2026-10-01
Open source