Calc AB · Unit 8 Progress Check

AP Calculus AB Unit 8 Progress Check Applications of Integration walkthrough.

AP Calculus AB Unit 8 is applications of integration, and the Unit 8 Progress Check MCQ Part A is mostly about setting up the right integral rather than evaluating it. This walkthrough covers the question types, the setup traps, and how to work through them. Explanations only — we never publish AP Classroom answer keys.

Updated August 2026Written by Mahmudul HasanFree · No signup

What Unit 8 covers

Unit 8 is Applications of Integration. Roughly 10–15% of the AP Calculus AB exam, and a reliable source of free-response points.

Average value of a function

(1/(b−a)) ∫_a^b f(x) dx — the average height, not the average of the endpoints.

Area between curves

∫ (top − bottom) dx, or (right − left) dy when integrating with respect to y.

Volumes by disk and washer

V = π ∫ R² dx, and the washer form π ∫ (R² − r²) dx when there is a hole.

Volumes with known cross-sections

Squares, semicircles, and triangles built on a base region.

Accumulation in context

Interpreting a definite integral of a rate as a net change, with units.

Motion

Position from velocity, and total distance as ∫ |v(t)| dt.

The Progress Check MCQ: what each question type tests

These are the question patterns that recur on this Progress Check, and what each one is really asking.

Choose the correct setup

Most questions ask for the integral, not its value. Identify the region and the variable of integration first.

dx or dy?

If horizontal slices are simpler, integrate with respect to y and use right minus left.

Disk vs washer

If the region touches the axis of rotation there is no hole — disk. If it does not, you need the washer.

Cross-section volumes

Write the side length as a function of x first, then apply the area formula for that shape.

Net change vs total distance

Displacement uses v(t); total distance uses |v(t)|. The MCQ tests the difference constantly.

The Progress Check FRQ

Unit 8 free response typically gives a region bounded by curves and asks for area, then a volume, then a rate interpretation. Points come from a correct integral setup with correct limits, correct orientation of the subtraction, and — on contextual parts — interpreting the integral in words with units.

Where students lose the most points

Squaring the difference instead of subtracting squares

The washer is π∫(R² − r²), never π∫(R − r)². This is the single most common Unit 8 error.

Reversing top and bottom

Area needs top minus bottom. A negative answer usually means the orientation flipped.

Wrong limits after switching variables

If you integrate with respect to y, the limits must be y-values.

Confusing displacement and distance

Total distance requires the absolute value of velocity.

Averaging endpoints

Average value is an integral divided by the interval width, not (f(a)+f(b))/2.

How to work through this unit

Sketch the region and draw one representative slice before writing anything. The slice tells you the variable, the limits, and whether you need a disk or a washer.

Once you have finished the Progress Check, put your raw score into our AP Calculus AB Calculator to see roughly where that pace puts you on the 1–5 scale, and use the AP Calculus AB Review for the full exam format and study plan.

Frequently asked questions

Quick answers — written by humans, not a chatbot.

What does the AP Calculus AB Unit 8 Progress Check cover?

Applications of integration: average value, area between curves, volumes by disk and washer, volumes with known cross-sections, accumulation in context, and motion.

What is the difference between the disk and washer method?

Use the disk method when the region touches the axis of rotation, and the washer method when there is a gap, subtracting the inner radius squared from the outer radius squared.

How do I find total distance travelled from velocity?

Integrate the absolute value of velocity. Integrating velocity alone gives displacement, which can be smaller or even zero.

Do you post AP Calculus Unit 8 answers?

No. We publish walkthroughs of the reasoning and the setup traps rather than AP Classroom answer keys.