Applications of Integration

What this unit covers

The topics below follow the published Calculus BC course framework for Unit 8. This unit is worth 6–9% 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 an integral for the area between two curves · Determine which function is on top and find the bounds from intersections · Choose between integrating with respect to x or y
• 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 · Relate average value to the Mean Value Theorem for Integrals
• Volumes of Solids15 min · 3 objectivesCompute volumes by the disk and washer methods · Set up volumes of solids with known cross sections · Select the correct method from the geometry of the region
• Arc Length13 min · 3 objectivesCompute the length of a curve using the arc-length integral · Set up arc length for a function y = f(x) · Recognize arc length as an accumulation of tiny hypotenuses

Formulas in Unit 8

Every term in Unit 8

All 10 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.

Arc length of y = f(x)

L = ∫√(1 + (dy/dx)²)dx over the interval. The integrand is almost never elementary, so these are usually calculator questions.

Arc length of a parametric curve

L = ∫√((dx/dt)² + (dy/dt)²)dt, which reduces to the Cartesian form when the parameter is x.

Volume by known cross sections

V = ∫A(x)dx with A(x) the cross-sectional area. Include π only if the cross sections are circular.

Average value of a function

f_avg = (1/(b − a))∫ₐᵇ f(x)dx — the height of the rectangle with the same area over the interval.

Total distance from a velocity function

∫|v(t)|dt, which requires splitting at every t where v changes sign. Displacement is ∫v(t)dt without absolute value.

Area between two curves

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

Volume by disks and washers

π∫r²dx for disks, π∫(R² − r²)dx for washers. Subtract the squares, never square the difference.

Choosing dx or dy

Integrate perpendicular to the slices — usually dx about a horizontal axis and dy about a vertical one.

Cross-section area formulas

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

Displacement vs total distance

Displacement is ∫v dt; total distance is ∫|v|dt, requiring a split wherever v changes sign.

What examiners penalize here

• When the region is easier to slice horizontally — for instance bounded by curves given as x = h(y) — integrate with respect to y using right minus left. Choosing the orientation that avoids splitting the region saves time on free response.
• Read the wording precisely: "average value of the velocity" means integrate v and divide by the time interval, while "average acceleration" means [v(b) − v(a)]/(b − a). One is an integral average, the other an endpoint average — pick the right formula.
• For cross-section problems, first decide whether the region’s height gives the *side* or the *diameter* of the shape. Semicircle and some square-on-the-diagonal setups use the height as a diameter, which changes the area formula — misreading this is a common lost point.
• Most arc-length integrals cannot be evaluated by hand, so on the calculator section, set up ∫√(1 + (dy/dx)²) dx precisely and let the calculator finish. The setup — correct derivative, correct bounds — is what the rubric rewards.

Practice Calculus BC

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 10 units of AP Calculus BC

1. Unit 1 · Limits & Continuity
2. Unit 2 · Differentiation: Definition & Fundamental Properties
3. Unit 3 · Differentiation: Composite, Implicit & Inverse Functions
4. Unit 4 · Contextual Applications of Differentiation
5. Unit 5 · Analytical Applications of Differentiation
6. Unit 6 · Integration & Accumulation of Change
7. Unit 7 · Differential Equations
8. Unit 8 · Applications of Integration
9. Unit 9 · Parametric, Polar & Vector-Valued Functions
10. Unit 10 · Infinite Sequences & Series