AP Calculus BC Unit 8 Applications of Integration: Area and Volume via Slicing

Area between curves

The area of a planar region bounded by curves (and usually endpoints), computed by summing thin slices and taking a definite integral.

Accumulation (idea)

Building a total (area or volume) by adding many small contributions (thin slices) and using a limit, which becomes a definite integral.

Definite integral

A limit of Riemann sums that gives the exact accumulated total over an interval (e.g., total area or total volume).

Vertical slice

A thin rectangle-like strip taken with thickness dx (slicing in the x-direction), used when integrating with respect to x.

Horizontal slice

A thin rectangle-like strip taken with thickness dy (slicing in the y-direction), used when integrating with respect to y.

Top minus bottom rule

For vertical slices, area is A = ∫_a^b (f(x) − g(x)) dx where f is the upper curve and g is the lower curve on the interval.

Right minus left rule

For horizontal slices, area is A = ∫_c^d (r(y) − ℓ(y)) dy where r is the right boundary and ℓ is the left boundary.

Bounds (limits of integration)

The start and end values (a to b or c to d) for an integral, typically found from intersections of boundary curves or given endpoints.

Intersection points

Points where boundary curves meet (found by solving equations like f(x)=g(x)); these often determine the integration limits.

Piecewise integral (splitting the interval)

Writing the area/volume as a sum of integrals over subintervals when the “top/bottom” (or “right/left”) relationship changes.

Signed area

The value of ∫(f−g) that can be negative on parts of an interval; it can cancel and fail to represent geometric area unless handled carefully.

Absolute value area form

A conceptual area setup A = ∫_a^b |f(x)−g(x)| dx; often replaced in practice by splitting into subintervals to avoid errors.

Base region

The 2D region in the plane that serves as the footprint of a 3D solid in volume-by-cross-sections problems.

Cross section

A slice of a solid made by a plane (often perpendicular to an axis); its shape and area determine the volume contribution.

Cross-sectional area function A(x) or A(y)

A formula giving the area of each slice as a function of position; volume is found by integrating this function.

Volume by known cross sections

A method where V = ∫ A(x) dx (or ∫ A(y) dy), using the area of each cross section based on a given shape.

Slice thickness (dx or dy)

The small dimension of each slice along the integration direction; it indicates whether you integrate with respect to x or y.

Solid of revolution

A 3D solid formed by rotating a plane region around a line (the axis of rotation).

Axis of rotation

The line a region is rotated around (e.g., x-axis, y-axis, y=k, x=h); radii are measured as distances to this line.

Disc method

Volume method for solids of revolution when slices have no hole: V = ∫ π(R(x))^2 dx (or with respect to y).

Washer method

Volume method for solids of revolution when slices have a hole: V = ∫ π((R(x))^2 − (r(x))^2) dx (or with respect to y).

Outer radius (R)

In the washer method, the larger distance from the axis of rotation to the farther boundary of the region.

Inner radius (r)

In the washer method, the smaller distance from the axis of rotation to the nearer boundary of the region (creates the hole).

Perpendicular-to-axis slicing rule (discs/washers)

Discs and washers use slices perpendicular to the axis of rotation: horizontal axis ⇒ use dx; vertical axis ⇒ use dy.

Symmetry for even integrands

If the integrand is even, ∫{-a}^{a} f = 2∫{0}^{a} f, often simplifying area/volume computations.