AP Calculus AB and BC

Unit 8: Applications of Integration

Exam weighting: AB 10-15% · BC 5-10%

Unit 8 turns the definite integral into a tool: integrate a rate to get accumulated change, or add up thin slices to get area and volume. It covers average value, motion, accumulation, area between curves, volumes by cross section, and solids of revolution, and it powers 10-15% of the AB exam.

The one move behind the whole unit

Every problem in Unit 8 is the same move: cut the quantity into thin pieces, write down the area, volume, or change of a single piece, then integrate to add them up. The skill the exam actually rewards is method selection, looking at a problem and knowing what one slice looks like before you write a single integral.

The topics fall into three strands. Topics 8.1 to 8.3 accumulate a rate: average value, motion, and accumulation functions each integrate something that changes over an interval. Topics 8.4 to 8.12 accumulate geometry: area between curves, then volumes built from cross sections, then volumes of revolution by the disc and washer methods. Topic 8.13, arc length, accumulates length and is BC only. Every strand is the limit-of-a-Riemann-sum idea from Unit 6 applied to a new quantity.

average value=1b−a∫abf(x) dxdisplacement=∫abv(t) dt,distance=∫ab∣v(t)∣ dtarea between curves=∫ab(top−bottom) dxvolume by cross sections=∫abA(x) dxdisc, then washer=π∫abR2 dx,π∫ab(R2−r2) dx\begin{aligned} \text{average value} &= \frac{1}{b-a}\int_a^b f(x)\,dx \\[4pt] \text{displacement} &= \int_a^b v(t)\,dt,\quad \text{distance}=\int_a^b |v(t)|\,dt \\[4pt] \text{area between curves} &= \int_a^b \big(\text{top}-\text{bottom}\big)\,dx \\[4pt] \text{volume by cross sections} &= \int_a^b A(x)\,dx \\[4pt] \text{disc, then washer} &= \pi\int_a^b R^2\,dx,\quad \pi\int_a^b (R^2-r^2)\,dx \end{aligned}
The problem gives youRecognize it asTopics
A function's average value over an intervalAverage value =1b−a∫abf(x) dx=\frac{1}{b-a}\int_a^b f(x)\,dx8.1
A rate of change and asks for a total or net amountNet change =∫abr(t) dt=\int_a^b r(t)\,dt; total =f(a)+∫abr(t) dt= f(a)+\int_a^b r(t)\,dt8.3
Velocity, and asks how far the particle actually travelsTotal distance =∫ab∣v(t)∣ dt= \int_a^b |v(t)|\,dt8.2
Two curves bounding a regionIntegrate top minus bottom in xx, or right minus left in yy8.4 to 8.6
A base region with a named cross-section shapeIntegrate that shape's area A(x)A(x) along the base8.7, 8.8
A region revolved around a line, touching the axisDisc: π∫abR2 dx\pi\int_a^b R^2\,dx8.9, 8.10
A region revolved with a gap to the axisWasher: π∫ab(R2−r2) dx\pi\int_a^b (R^2-r^2)\,dx8.11, 8.12

Topic 8.1 asks for average value, which students mix up with average rate of change: average value divides an integral by the width of the interval, while average rate of change divides a change in ff by a change in xx. Topic 8.2 hides the exam's favorite trap: the integral of velocity v(t)v(t) gives displacement, but total distance needs the integral of speed ∣v(t)∣|v(t)|, so you find where v(t)v(t) changes sign and split the interval. Topic 8.3 problems hand you a rate and ask for a total: the net change is ∫abr(t) dt\int_a^b r(t)\,dt, and the total amount is the starting value plus that integral.

For area (8.4 to 8.6), sketch the region and choose between integrating in xx or in yy by which one gives a clean top minus bottom (or right minus left) without carving the region into extra pieces; when curves cross at more than two points (8.6), split at each crossing or integrate the absolute value of the difference. For volume, first separate cross-section solids (8.7, 8.8), where you integrate the area of a named shape sitting on a base, from solids of revolution (8.9 to 8.12). Within revolution, run the gap test: if the region touches the axis, use a disc; if a gap sits between the region and the axis, use a washer. When the axis is a line other than the x- or y-axis (8.10, 8.12), write each radius as the distance from the curve to that line, not just the function value.

Unit 8 is 10 to 15% of the AB exam and 6 to 9% of BC, and it pulls more than its weight on the free-response section. Expect a calculator-active accumulation question where you set up an integral exactly and evaluate it numerically, plus an area-and-volume question that usually revolves a region around some line. Multiple-choice items lean on average value, total distance, and volume setups. On the non-calculator section you antidifferentiate by hand, so the setup and the integration both earn points.

Topics in this unit

Topic numbers and titles from the College Board Course and Exam Description.

  • 8.1Finding the Average Value of a Function on an Interval
  • 8.2Connecting Position, Velocity, and Acceleration of Functions Using Integrals
  • 8.3Using Accumulation Functions and Definite Integrals in Applied Contexts
  • 8.4Finding the Area Between Curves Expressed as Functions of x
  • 8.5Finding the Area Between Curves Expressed as Functions of y
  • 8.6Finding the Area Between Curves That Intersect at More Than Two Points
  • 8.7Volumes with Cross Sections: Squares and Rectangles
  • 8.8Volumes with Cross Sections: Triangles and Semicircles
  • 8.9Volume with Disc Method: Revolving Around the x- or y-Axis
  • 8.10Volume with Disc Method: Revolving Around Other Axes
  • 8.11Volume with Washer Method: Revolving Around the x- or y-Axis
  • 8.12Volume with Washer Method: Revolving Around Other Axes
  • 8.13The Arc Length of a Smooth, Planar Curve and Distance TraveledBC only

How to study this unit

  • Topic 8.2 is the classic trap: read whether the question wants displacement or total distance. Displacement integrates $v(t)$; total distance integrates $|v(t)|$, which means finding where $v(t)$ changes sign and splitting the integral there.
  • For area (8.4, 8.5), always sketch first, then pick $dx$ or $dy$ by whichever gives one clean top-minus-bottom integral. If the curves cross more than twice (8.6), split at every intersection or integrate the absolute value of the difference.
  • For solids of revolution (8.9 to 8.12), run the gap test: a region touching the axis means disc, a gap means washer. When you revolve around a line that is not the x- or y-axis (8.10, 8.12), rebuild each radius as the distance from the curve to that line before squaring.
  • Treat Topic 8.3 accumulation problems as calculator-active: set the integral up exactly and evaluate it numerically. The net change is the integral of the rate; the total amount is the starting value plus that integral, so do not forget to add the initial amount.
  • Keep average value (8.1) separate from average rate of change: average value divides the integral by $(b-a)$, average rate of change divides the change in $f$ by the change in $x$. BC students, note that arc length (8.13) is the only BC-only topic in this unit.

Guides for this unit

Tools and tables for this unit