Applications of Integration

What this unit covers

The topics below follow the published Calculus AB course framework for Unit 8. This unit is worth 10–15% of the exam, so budget your time against that rather than against how long the unit takes to teach.

Lessons in this unit

• Area Between Curves14 min · 3 objectivesSet up the integral of (top − bottom) for the area between two curves · Find intersection points to determine the limits of integration · Integrate with respect to y when curves are functions of y
• Volumes: Disks, Washers & Cross Sections15 min · 3 objectivesCompute volumes of revolution with the disk method · Use the washer method when the solid has a hole · Find volumes of solids with known cross-sectional shapes
• Average Value of a Function12 min · 3 objectivesCompute the average value of a function over an interval · Distinguish the average value of f from the average rate of change · Interpret the average value geometrically and in context
• Motion Revisited: Accumulating Velocity14 min · 3 objectivesRecover displacement by integrating velocity over an interval · Compute total distance traveled by integrating the speed |v(t)| · Find position from velocity and an initial condition

Formulas in Unit 8

Every term in Unit 8

All 13 terms we publish for Applications of Integration, with definitions. Reading them through is the fastest way to find the ones you cannot define — then drill those in cram mode until you can produce them without the prompt.

Average value of a function

f_avg = (1/(b − a))∫ₐᵇ f(x)dx. Not the average of the endpoints, and not the average of the derivative.

Mean Value Theorem for integrals

Some c in [a, b] has f(c) equal to the average value, provided f is continuous.

Area between two curves

∫(top − bottom)dx, splitting at every intersection where the curves swap position.

Integrating with respect to y

When the curves are more naturally functions of y, use ∫(right − left)dy. Often avoids splitting the region.

Volume by disks

V = π∫r² dx, where r is the distance from the axis to the curve. Used when the region touches the axis of rotation.

Volume by washers

V = π∫(R² − r²)dx with outer and inner radii. Squaring the difference instead of subtracting the squares is the standard error.

Volume by known cross sections

V = ∫A(x)dx, where A(x) is the area of the cross section. No π unless the cross sections are circular.

Cross-section area formulas

Square s², equilateral triangle (√3/4)s², semicircle (π/8)s² where s is the diameter. Identify whether s is a side or a diameter.

Choosing the variable of integration

Integrate perpendicular to the slices. Rotating about a horizontal axis usually means dx; a vertical axis usually means dy.

Setting up an area integral

Sketch first, find the intersections, and decide which curve is on top on each subinterval before writing anything down.

Washers with a shifted axis

When rotating about y = k rather than the x-axis, the radii are |f(x) − k|, not f(x). Forgetting this is the most common volume error.

Interpreting average value

The constant value that would produce the same accumulated total over the interval — state it with units in context.

Choosing disks or washers

Disks when the region touches the axis of rotation, washers when there is a gap between the region and the axis.

What examiners penalize here

• Before integrating, sketch the two curves and their intersections to see which is on top and whether they cross inside the interval. That quick picture prevents both limit errors and top/bottom mix-ups that graders penalize.
• Identify the cross-section shape before writing the integral: disks and washers bring a π (they are circles), but squares, triangles, and rectangles do not. Slapping a π on a non-circular cross section is a classic AP volume error.
• When a free-response question asks for the "average value" of a quantity, use the integral formula and divide by the interval length. If it asks for the "average rate of change", use the endpoint difference over the length instead — the two prompts want different computations.
• Watch the wording: "displacement" or "position change" means ∫v dt, while "total distance traveled" means ∫|v| dt with the interval split at sign changes. And "where is the particle" always needs the initial position added to the integral.

Practice Calculus AB

Our practice bank is drawn from across the whole course rather than filtered to one unit, which is closer to how the exam asks anyway — it will not tell you which unit a question is testing.

Questions about this unit

All 8 units of AP Calculus AB

1. Unit 1 · Limits & Continuity
2. Unit 2 · Differentiation: Definition & Rules
3. Unit 3 · Composite & Implicit Differentiation
4. Unit 4 · Contextual Applications
5. Unit 5 · Analytical Applications
6. Unit 6 · Integration & Accumulation
7. Unit 7 · Differential Equations
8. Unit 8 · Applications of Integration