Put the integral to work on geometry and motion. Compute average values, connect position and velocity with integrals, find areas between curves, and build volumes with cross sections, discs, and washers — plus arc length for BC.
All 13 topics from the College Board CED, in order. Topic 8.13 is BC only.
Topic 8.1
Average Value of a Function
The mean value of f over [a, b] via integration.
Topic 8.2
Motion with Integrals
Position, velocity, and acceleration connected by integrals.
Topic 8.3
Accumulation in Applied Contexts
Definite integrals modeling accumulated change.
Topic 8.4
Area Between Curves (in x)
Integrating top minus bottom with respect to x.
Topic 8.5
Area Between Curves (in y)
Integrating right minus left with respect to y.
Topic 8.6
Area with Multiple Intersections
Splitting the region where curves cross repeatedly.
Topic 8.7
Cross Sections: Squares & Rectangles
Volumes built from square/rectangular slices.
Topic 8.8
Cross Sections: Triangles & Semicircles
Volumes from triangular/semicircular slices.
Topic 8.9
Disc Method (x- or y-axis)
Revolving a region into solid discs.
Topic 8.10
Disc Method (Other Axes)
Discs revolved about a shifted axis.
Topic 8.11
Washer Method (x- or y-axis)
Revolving a region with a hole.
Topic 8.12
Washer Method (Other Axes)
Washers revolved about a shifted axis.
Topic 8.13
Arc Length & Distance (BC)
Length of a smooth curve and total distance.
About Unit 8
Unit 8 applies the definite integral to geometry and motion. You compute the average value of a function, (1/(b−a))∫₀ᵇ f dx, connect position, velocity, and acceleration through integrals (displacement = ∫v dt, total distance = ∫|v| dt), and use accumulation integrals in applied contexts.
The geometric heart of the unit is area and volume. Find the area between curves by integrating top − bottom (in x) or right − left (in y), splitting when curves cross. Build volumes with known cross sections (squares, rectangles, triangles, semicircles), and revolve regions into solids with the disc method (V = π∫R² dx) and the washer method (V = π∫(R² − r²) dx), including revolutions about shifted axes. BC students add arc length, ∫√(1 + (dy/dx)²) dx.
On the exam this unit is 10–15% for AB and 6–9% for BC, and takes about 19 class periods (AB). The four mathematical practices below run through every topic: