Applications of Integration

What this unit covers

The topics below follow the published Calculus AB course framework for Unit 8. This unit is worth 10–15% 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 the integral of (top − bottom) for the area between two curves · Find intersection points to determine the limits of integration · Integrate with respect to y when curves are functions of y
• Volumes: Disks, Washers & Cross Sections15 min · 3 objectivesCompute volumes of revolution with the disk method · Use the washer method when the solid has a hole · Find volumes of solids with known cross-sectional shapes
• 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 · Interpret the average value geometrically and in context
• Motion Revisited: Accumulating Velocity14 min · 3 objectivesRecover displacement by integrating velocity over an interval · Compute total distance traveled by integrating the speed |v(t)| · Find position from velocity and an initial condition
• Choosing dx or dy, and Splitting When You Must15 min · 3 objectivesDecide between vertical and horizontal slices for an area problem · Set up an area integral requiring more than one piece · Find the limits of integration by locating intersections
• Net Change: What an Integral of a Rate Means14 min · 3 objectivesState the Net Change Theorem and identify when a problem calls for it · Compute a final amount from an initial amount and a rate · Set up an in-minus-out problem and find when the amount is greatest
• Volumes with Known Cross Sections15 min · 3 objectivesSet up a volume integral for solids with square, rectangular, triangular, or semicircular cross sections · Identify the base region and express the cross-sectional side in terms of it · Distinguish cross-section volumes from disk and washer revolutions

Formulas in Unit 8

Every term in Unit 8

All 39 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.

Average value of a function

f_avg = (1/(b − a))∫ₐᵇ f(x)dx. Not the average of the endpoints, and not the average of the derivative.

Mean Value Theorem for integrals

Some c in [a, b] has f(c) equal to the average value, provided f is continuous.

Area between two curves

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

Integrating with respect to y

When the curves are more naturally functions of y, use ∫(right − left)dy. Often avoids splitting the region.

Volume by disks

V = π∫r² dx, where r is the distance from the axis to the curve. Used when the region touches the axis of rotation.

Volume by washers

V = π∫(R² − r²)dx with outer and inner radii. Squaring the difference instead of subtracting the squares is the standard error.

Volume by known cross sections

V = ∫A(x)dx, where A(x) is the area of the cross section. No π unless the cross sections are circular.

Cross-section area formulas

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

Choosing the variable of integration

Integrate perpendicular to the slices. Rotating about a horizontal axis usually means dx; a vertical axis usually means dy.

Setting up an area integral

Sketch first, find the intersections, and decide which curve is on top on each subinterval before writing anything down.

Washers with a shifted axis

When rotating about y = k rather than the x-axis, the radii are |f(x) − k|, not f(x). Forgetting this is the most common volume error.

Interpreting average value

The constant value that would produce the same accumulated total over the interval — state it with units in context.

Choosing disks or washers

Disks when the region touches the axis of rotation, washers when there is a gap between the region and the axis.

Vertical slices mean top minus bottom

Integrating dx means each slice runs from the lower curve to the upper one, both written as functions of x.

Horizontal slices mean right minus left

Integrating dy means each slice runs from the left curve to the right one, both written as functions of y.

Switch variables before splitting

Two integrals doubles the chance of an algebra error. Setting up in the other variable usually costs only solving for the inverse.

Test which curve is on top

Substitute a point inside the interval rather than judging from the equations. A negative area means you reversed them.

Limits come from intersections

Solve the boundary equations simultaneously first. If integrating dy, the limits are y values, which may need substituting back.

More than two intersections

The upper curve can switch, in which case the region genuinely requires splitting at the crossing. The exam includes this case.

Keep decimals until the end

A calculator intersection like 1.8637 should be carried at three or more places; rounding early moves the answer outside tolerance.

Volume by cross sections, general form

V = ∫A(x)dx, where A(x) is the cross-sectional area. Disks and washers are special cases, not separate topics.

Cross-sectional side from the base

For slices perpendicular to the x-axis, s = f(x) − g(x) — the same top-minus-bottom expression used for area.

Square cross section

A = s², where s is the side taken from the base region. The simplest case, and the one that makes a √x base region come out clean.

Equilateral triangle cross section

A = (√3/4)s². The factor √3/4 ≈ 0.433 comes from the height (√3/2)s, and it is the one shape factor worth memorizing outright.

Semicircle on diameter s

A = (π/8)s², because the radius is s/2. Using πs²/2 is the most frequent error in this topic.

Rectangle cross section with a stated ratio

A = s × (ratio × s). A height twice the base gives 2s².

Perpendicular to which axis

Perpendicular to the x-axis means vertical slices and dx; perpendicular to the y-axis means horizontal slices and dy. Decide before setting up.

Washer inner and outer radii

Both measured from the axis of rotation, so a shifted axis changes both. Compute each radius as a distance, not as a function value.

Net change from a rate in an applied context

∫ₐᵇ f′(t)dt = f(b) − f(a). The integral of a rate is the net change in the quantity.

Amount equals initial plus change

f(b) = f(a) + ∫ₐᵇ f′(t)dt. Reporting the integral alone as the amount is the most common applied-integration error.

In minus out

The net rate is R(t) − S(t), and the amount is the initial value plus the integral of that difference.

When the amount is greatest

Where the net rate changes from positive to negative, that is where R = S with inflow dominating before. Then compare with the endpoints.

Average value formula

(1/(b − a))∫ₐᵇ f dx. The division by the interval width is what turns a total into an average.

Average value versus average rate of change

Average value integrates f; average rate of change is [f(b) − f(a)]/(b − a) and involves no integral. Check which function you were given.

Average value of a rate keeps the rate’s units

The average of a velocity in feet per second is a velocity, not a distance.

MVT for integrals

A continuous function attains its average value somewhere on the interval — the integral analogue of the Mean Value Theorem.

Area is never negative

If an area computation returns a negative number, the curves were subtracted in the wrong order.

Set up before computing

Free-response rubrics award the correct integral expression separately from the numerical answer, so write the integral even if the arithmetic defeats you.

Volume of revolution about a horizontal line

Radius is the vertical distance from the curve to that line, so it is |f(x) − k| rather than f(x).

What examiners penalize here

• Before integrating, sketch the two curves and their intersections to see which is on top and whether they cross inside the interval. That quick picture prevents both limit errors and top/bottom mix-ups that graders penalize.
• Identify the cross-section shape before writing the integral: disks and washers bring a π (they are circles), but squares, triangles, and rectangles do not. Slapping a π on a non-circular cross section is a classic AP volume error.
• When a free-response question asks for the "average value" of a quantity, use the integral formula and divide by the interval length. If it asks for the "average rate of change", use the endpoint difference over the length instead — the two prompts want different computations.
• Watch the wording: "displacement" or "position change" means ∫v dt, while "total distance traveled" means ∫|v| dt with the interval split at sign changes. And "where is the particle" always needs the initial position added to the integral.
• Always determine **which curve is on top** by testing a point inside the interval, not by looking at the equations. Reversing top and bottom produces the negative of the right answer, and an area cannot be negative — so if your result comes out negative, that is the error.

Practice Calculus AB

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 8 units of AP Calculus AB

1. Unit 1 · Limits & Continuity
2. Unit 2 · Differentiation: Definition & Rules
3. Unit 3 · Composite & Implicit Differentiation
4. Unit 4 · Contextual Applications
5. Unit 5 · Analytical Applications
6. Unit 6 · Integration & Accumulation
7. Unit 7 · Differential Equations
8. Unit 8 · Applications of Integration