Unit 8 Applications of Integration (AP Calculus BC): Average Value, Net Change & Motion, Area Between Curves, and Volume

Average value of a function

A single “typical” value of f(x) on [a,b], defined by favg = (1/(b−a))∫a^b f(x) dx.

Average value formula

favg = (1/(b−a))∫a^b f(x) dx; “add” via the integral, then divide by interval length.

Signed area

The value of ∫_a^b f(x) dx, counting area above the x-axis as positive and below as negative.

Mean height interpretation (average value)

f_avg is the constant height of a rectangle over [a,b] having the same signed area as the region under y=f(x).

Average rate interpretation (average value)

If f is a rate, then ∫_a^b f gives total accumulated amount, and dividing by (b−a) gives the average rate.

Common average value misconception

The average value f_avg does not have to equal f(a) or f(b); it depends on values over the whole interval.

Mean Value Theorem for Integrals (MVTI)

If f is continuous on [a,b], then there exists c in [a,b] such that f(c) = (1/(b−a))∫_a^b f(x) dx.

Value c that attains the average value

A point c in [a,b] guaranteed (when f is continuous) to satisfy f(c)=f_avg.

“Set up an equation for c” (MVTI)

To find c, compute favg and solve f(c)=favg, keeping only solutions within [a,b].

Net change

The total change in a quantity over an interval, computed as ∫ (rate of change) = final value − initial value.

Net Change Theorem (Integral of a rate)

If F′(t)=r(t), then ∫_a^b r(t) dt = F(b) − F(a).

Fundamental Theorem of Calculus (FTC) connection

Integrating a derivative over [a,b] gives the original function’s change: ∫_a^b F′(t) dt = F(b) − F(a).

Position function s(t)

A function giving location along a line as a function of time t.

Velocity v(t)

The derivative of position: v(t)=s′(t).

Acceleration a(t)

The derivative of velocity (and second derivative of position): a(t)=v′(t)=s′′(t).

Displacement

Net change in position: s(b)−s(a)=∫_a^b v(t) dt.

Total distance traveled

Total path length in 1D motion: ∫_a^b |v(t)| dt (movement regardless of direction).

Displacement vs. distance trap

∫ v(t) dt gives displacement (can cancel); distance requires |v(t)| and often splitting where v(t)=0.

Direction change time

A time when v(t)=0 and the sign of v changes, indicating the particle reverses direction.

Splitting an integral for distance

Breaking ∫_a^b |v(t)| dt into pieces at times where v(t)=0 so the absolute value can be handled correctly.

Accumulation from a rate (units idea)

Integrating a “per time” rate over time produces a total amount; units help interpret ∫ r(t) dt.

Initial condition (in motion/accumulation)

A starting value like s(a) or v(a) needed to determine an absolute position/velocity after integrating.

Area between curves

Geometric area enclosed by two graphs, computed by integrating the difference between the top and bottom functions (or right minus left).

Top minus bottom (vertical slices)

For area with respect to x: A = ∫_a^b (f(x)−g(x)) dx where f is above g on [a,b].

Intersection points (area bounds)

x-values (or y-values) where the curves meet, found by solving f=g, often used as integration limits.

When to split an area-between-curves integral

If the curves cross within the interval (the “top” function changes), split the integral to prevent cancellation.

Horizontal slices (integrating with respect to y)

Area setup A = ∫_c^d (R(y)−L(y)) dy, using right boundary minus left boundary.

Right boundary R(y)

For horizontal slices, the x-value of the rightmost curve expressed as a function of y.

Left boundary L(y)

For horizontal slices, the x-value of the leftmost curve expressed as a function of y.

Cross-sectional volume method (known slices)

Volume found by summing cross-sectional areas: V=∫_a^b A(x) dx (or ∫ A(y) dy depending on orientation).

Cross-sectional area function A(x)

The area of a slice perpendicular to the x-axis at position x; used inside V=∫ A(x) dx.

“Perpendicular to the x-axis” (slicing meaning)

Slices are vertical, thickness dx, so integrals are typically in terms of x.

“Perpendicular to the y-axis” (slicing meaning)

Slices are horizontal, thickness dy, so integrals are typically in terms of y.

Square cross-sections

If side length is s(x), then cross-sectional area is A(x)=s(x)^2.

Semicircle cross-sections

If radius is r(x), then cross-sectional area is A(x)=(1/2)πr(x)^2 (watch diameter vs radius).

Diameter vs. radius mistake (semicircles)

If the diameter is d, the radius is d/2; using d as r causes the area to be 4 times too large.

Solid of revolution

A 3D solid formed by rotating a 2D region around a line (axis of rotation).

Axis of rotation

The line a region is rotated about (e.g., x-axis, y-axis, y=k, or x=k), which determines radii distances.

Disk method

Volume method for rotation with no hole: V=∫_a^b πR(x)^2 dx (or with respect to y).

Washer method

Volume method for rotation with a hole: V=∫_a^b π(R(x)^2−r(x)^2) dx; subtract areas, not radii.

Outer radius R

In washers, the larger distance from the axis of rotation to the outer curve.

Inner radius r

In washers, the smaller distance from the axis of rotation to the inner curve (creates the hole).

Washer formula common error

Writing π(R−r)^2 is incorrect; the correct expression is π(R^2−r^2).

Choosing dx vs dy for disks/washers

Disks/washers use slices perpendicular to the axis: horizontal axis → often dx; vertical axis → often dy.

Cylindrical shell method

Volume method using slices parallel to the axis of rotation: V=∫ 2π(radius)(height) dx (or dy).

Shell radius

Distance from the shell to the axis of rotation (often requires absolute value for shifted axes like x=2).

Shell height

Length of the shell parallel to the axis, typically “top minus bottom” of the region at that x (or y).

Absolute value in shell radius

When rotating about x=k (or y=k), radius is a distance like |x−k|, which may require splitting the integral.

Method-choice heuristic (modeling workflow)

Identify what’s accumulated, choose a slicing direction, express slice measurements, find bounds, then integrate (instead of guessing formulas).

“Set up but do not evaluate” integral

An AP-style prompt focusing on correct modeling: write the correct integral expression with correct radii/heights/bounds without computing it.