AP Calculus AB and BC · Exam guide
AP Calculus Area and Volume FRQ: Washers, Disks, and Cross Sections
Area and volume questions reward a clean setup. The integral is the answer, and the grader checks the limits, the integrand, and the form (area, washer, or cross section) before the number. This guide shows the three common setups with checked arithmetic.
Area between curves
For \(f(x) \ge g(x)\) on \([a, b]\), the area is \(\int_a^b (f(x) - g(x))\,dx\). Find the limits where the curves cross, and decide which curve is on top.
Let \(R\) be the region between \(f(x) = 2x - x^2\) and \(g(x) = x\). Find the area of \(R\).
Set \(2x - x^2 = x\): \(x - x^2 = 0\), so the curves meet at \(x = 0\) and \(x = 1\). On \([0, 1]\), \(f(x) \ge g(x)\). Area \(= \int_0^1 (x - x^2)\,dx = \left[\dfrac{x^2}{2} - \dfrac{x^3}{3}\right]_0^1 = \dfrac{1}{6}\).
Volume by washers
The region \(R\) is revolved about the \(x\)-axis. Find the volume.
The outer radius is \(f(x) = 2x - x^2\), and the inner radius is \(g(x) = x\). Volume: \[\pi\int_0^1 \left[(2x - x^2)^2 - x^2\right]dx = \pi\int_0^1 (3x^2 - 4x^3 + x^4)\,dx = \pi\left(1 - 1 + \dfrac{1}{5}\right) = \dfrac{\pi}{5}\]
It squares the difference. For washers, square each radius first and then subtract.
It squares each radius separately.
Volume by cross sections
The base of a solid is the region under \(y = \sqrt{x}\) for \(0 \le x \le 4\). Cross sections perpendicular to the \(x\)-axis are squares. Find the volume.
The side of each square is \(\sqrt{x}\), so its area is \(x\). Volume \(= \int_0^4 x\,dx = 8\).
What to write on the page
- The integral with limits and integrand, before any number.
- Which curve is on top, or which radius is outer, and how you know (a sentence or a labeled sketch).
- A final answer with units if the problem gives units.
- On Part A, an integral can be evaluated on the calculator, but the integral must be written first.
Common mistakes
- Using \(\pi\) when the solid is not a solid of revolution, such as a square cross section.
- Subtracting the radii before squaring them in a washer problem.
- Using the wrong axis of rotation, or not shifting the radius when the axis is a line such as \(y = -1\).
- Finding the wrong limits because the curves cross more than once.
Frequently asked questions
Do I have to simplify the integral before the calculator?
No. Write the integral as set up, then evaluate it.
How do I decide between disks and washers?
Use disks when the region touches the axis of rotation along its whole length, and washers when there is a gap.
What about shell method?
Shell method is not required on the exam. Washers and cross sections are the tested methods.
Do I need units on a volume?
Yes, if the problem gives units, such as cubic centimeters.
Where can I find the official scoring rules?
Exam formats change a little from year to year, so check the current course and exam description on the College Board site before test day.
Keep practicing
Every lesson in the AP Calculus courses ends with practice problems and an Exam focus box that shows how the topic is tested, and every unit has a 10-problem practice set with worked answers. These free lessons also build the skills the free-response section tests:
See the AP Calculus AB and BC course Free lessons and lesson list See the AP Calculus BC course
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Multiple choice
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