AP Calculus AB Unit 8 Notes: Finding Volumes Using Integration

Volume by Cross Sections

A method for finding the volume of a solid by slicing it into thin slabs perpendicular to an axis and integrating the cross-sectional area over an interval.

Cross-Sectional Area Function A(x)

The area of the slice (face) of the solid at position x; used in the volume integral V = ∫_a^b A(x) dx.

Cross-Section Volume Formula (dx)

If slices are perpendicular to the x-axis, the volume is V = ∫_a^b A(x) dx, where dx represents slice thickness in the x-direction.

Cross-Section Volume Formula (dy)

If slices are perpendicular to the y-axis, the volume is V = ∫_c^d A(y) dy, where dy represents slice thickness in the y-direction.

Slice Thickness (dx or dy)

The small measurement representing how “thick” each slab is in the slicing direction; becomes dx or dy in the integral limit process.

Base Region

The 2D region in the plane (bounded by curves/lines) that defines the footprint of the 3D solid used in cross-section problems.

Segment Length s(x) (Top Minus Bottom)

For vertical slices (perpendicular to the x-axis), s(x) = ytop(x) − ybottom(x), representing the vertical distance between bounding curves.

Segment Length s(y) (Right Minus Left)

For horizontal slices (perpendicular to the y-axis), s(y) = xright(y) − xleft(y), representing the horizontal distance between bounding curves.

Square Cross Section

A cross section whose face is a square; if side length is s(x), then area A(x) = (s(x))^2.

Rectangle Cross Section with Multiple k

A rectangle cross section where one side is s(x) and the other side is k·s(x); area A(x) = k(s(x))^2.

Right-Triangle Cross Section (Legs from Base Segment)

A cross section that is a right triangle where base segment length is s(x) and height is k·s(x); area A(x) = (k/2)(s(x))^2.

Equilateral Triangle Cross Section

A cross section that is an equilateral triangle with side length s(x); area A(x) = (√3/4)(s(x))^2.

Semicircle Cross Section

A cross section whose face is a semicircle; area is half of a circle’s area, A = (1/2)πr^2.

Semicircle Diameter-to-Radius Conversion

If the given base segment is the diameter s(x), then the radius is r(x) = s(x)/2 (a common required step).

Semicircle Area in Terms of Diameter s(x)

When s(x) is the diameter, A(x) = (1/2)π(s(x)/2)^2 = (π/8)(s(x))^2.

Even Integrand Symmetry

If f(x) is even, then ∫-a^a f(x) dx = 2∫0^a f(x) dx, often simplifying volume integrals (e.g., semicircle examples).

Disc Method

A rotational volume method where rotating a region that touches the axis produces solid discs; uses cross-sectional area πr^2.

Disc Method Volume Formula

For rotation about an axis with slices perpendicular to the axis: V = ∫ π(r(x))^2 dx (or V = ∫ π(r(y))^2 dy).

Radius as Distance to Axis of Rotation

In rotational problems, the radius is the distance from the axis of rotation to the curve/boundary (e.g., about y=k: r(x)=|f(x)−k|).

Rotation About a Shifted Line (y = k)

A rotation around a horizontal line y=k; radii are computed using distances to y=k, typically |f(x)−k|.

Washer Method

A rotational volume method used when the region does not touch the axis of rotation, creating a hole; cross sections look like washers (discs with holes).

Washer Area Formula

The area of a washer is A = πR^2 − πr^2, where R is the outer radius and r is the inner radius.

Washer Method Volume Formula

V = π∫a^b (R(x)^2 − r(x)^2) dx (or V = π∫c^d (R(y)^2 − r(y)^2) dy), depending on slice direction.

Outer Radius vs. Inner Radius (Washers)

Outer radius R is the farther distance from the axis to the region; inner radius r is the nearer distance—both measured as distances to the axis.

Common Setup Errors (Volumes)

Frequent mistakes include using wrong segment (top-bottom vs right-left), squaring incorrectly, using diameter instead of radius, swapping washer radii, and using mismatched bounds for dx vs dy.