AP Calculus AB/BC Unit 8 Cheat Sheet: Applications of Integration

The basics

What it covers: Average value, motion via integrals, area between curves, volumes by cross sections, the disc and washer methods, and (BC) arc length.

Exam weight: About 10–15% of the AB exam and 6–9% of the BC exam.

The big question: How do definite integrals measure areas, volumes, and total change?

Mathematical practices: Implementing Mathematical Processes, Connecting Representations, Justification, and Communication & Notation.

Key topics at a glance

Average Value

fⁱᵛᵍ = (1/(b−a))∫₀ᵇ f dx. The Mean Value Theorem for Integrals guarantees f attains it at some c.

Motion with Integrals

Displacement = ∫v dt; total distance = ∫|v| dt; s(b) = s(a) + ∫v dt; v(t) = v₀ + ∫a dt.

Area Between Curves

∫(top − bottom) dx (vertical slices) or ∫(right − left) dy (horizontal). Split where curves cross.

Cross-Section Volumes

V = ∫₀ᵇ A(x) dx. Square: A = s²; semicircle: A = (π/8)s²; where s = distance between curves.

Disc Method

V = π∫ R² dx — no hole; R is the distance from the axis to the curve.

Washer Method

V = π∫(R² − r²) dx — a hole; R outer, r inner radius.

Shifted Axes

Radius = distance to the axis: e.g. about y = k, R = |curve − k|. Adjust R and r accordingly.

Arc Length (BC)

L = ∫₀ᵇ √(1 + (dy/dx)²) dx for a smooth curve.

The key formulas you must know

Average value — (1/(b−a))∫₀ᵇ f dx.
Displacement — ∫v dt (signed).
Total distance — ∫|v| dt.
Area between curves (x) — ∫(top − bottom) dx.
Area between curves (y) — ∫(right − left) dy.
Cross-section volume — ∫₀ᵇ A(x) dx.
Disc method — π∫ R² dx.
Washer method — π∫(R² − r²) dx.
Radius — distance from the axis of revolution to the curve.
Arc length (BC) — ∫√(1 + (dy/dx)²) dx.

Key themes to remember

Integrate the right slice. Area = length of a slice; volume = area of a slice.
Displacement vs. distance: integrate v vs. |v|.
Match the variable to the slices. Vertical → dx; horizontal → dy.
Disc vs. washer: a gap from the axis means a hole (washer).
Radius is a distance. For shifted axes, measure to that axis.

Common exam traps

Divide by (b − a) for average value — don’t report just the integral.
Distance needs |v|. Integrating v gives displacement, not distance.
Top minus bottom, not the reverse — keep slice heights positive.
Split at every intersection when curves cross more than twice.
Square the radius, and for washers subtract r² from R² (not (R − r)²).
Shifted axis changes the radius — subtract the axis value.
Match dx to dy setups: revolving about the y-axis often needs functions of y.