AP Calculus AB Unit 8: Applications of Integration (Comprehensive Study Notes)

Average value of a function

The mean output level of f on [a,b], defined by favg = (1/(b−a))∫a^b f(x) dx.

Signed area

The value of ∫_a^b f(x) dx interpreted as area above the x-axis minus area below the x-axis.

Equal-area rectangle interpretation

A visualization where the average value is the height of a rectangle of width (b−a) having the same (signed) area as the region under f on [a,b].

Interval length (b−a)

The width of the x-interval used to convert total accumulation ∫_a^b f(x) dx into an average via division.

Integrable on [a,b]

A function for which the definite integral ∫_a^b f(x) dx exists (so average value can be computed).

Average rate of change

The secant slope from a to b: (f(b)−f(a))/(b−a), which is different from average value.

Secant slope

Slope of the line through (a,f(a)) and (b,f(b)); equals the average rate of change on [a,b].

Midpoint misconception (average value)

The common mistake of assuming f_avg = f((a+b)/2); this is not the definition and can be wrong.

Average velocity

The average value of velocity v(t) on [a,b]: vavg = (1/(b−a))∫a^b v(t) dt.

Average speed

The average value of |v(t)| on [a,b], reflecting total distance per time rather than signed motion.

Net change

Overall change combining increases and decreases; computed by integrating a rate over an interval.

Net change formula

If Q changes at rate Q′(t), then Q(b)−Q(a) = ∫_a^b Q′(t) dt.

Rate function

A derivative like Q′(t) that describes how a quantity changes per unit time (or per unit input).

Accumulation function

A function defined by an integral with variable upper limit, e.g., A(x)=∫_a^x f(t) dt.

Variable upper limit

The upper bound x in an integral like ∫_a^x f(t) dt that makes the integral define a new function of x.

Fundamental Theorem of Calculus (FTC) for accumulation

If A(x)=∫_a^x f(t) dt and f is continuous, then A′(x)=f(x).

Position function s(t)

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

Velocity v(t)

The derivative of position: v(t)=s′(t), representing rate of change of position.

Acceleration a(t)

The derivative of velocity: a(t)=v′(t)=s″(t), representing rate of change of velocity.

Displacement

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

Distance traveled (total)

Total motion regardless of direction: ∫_a^b |v(t)| dt.

Change in velocity

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

Sign change (for |v| integration)

A point where v(t)=0 and velocity can switch sign, requiring the interval to be split when computing distance.

Piecewise integration for distance

Computing ∫|v(t)| by splitting at zeros of v(t) and integrating with sign corrections on each subinterval.

Absolute value pitfall (distance)

The mistake of using |∫a^b v(t) dt| instead of ∫a^b |v(t)| dt.

Units check (integration)

Using units to verify correctness: integrating a rate (units per time) over time gives the accumulated units.

Area between curves

The area of a region bounded by two graphs, computed by integrating the distance between them.

Vertical slicing

Area method using dx; rectangle height is (top function − bottom function).

Horizontal slicing

Area method using dy; rectangle width is (right function − left function).

Top minus bottom

For vertical slices, the integrand for area: f(x)−g(x), where f is above g on the interval.

Right minus left

For horizontal slices, the integrand for area: R(y)−L(y), where R is to the right of L.

Intersection points

Values where bounding curves meet (found by setting them equal), used as bounds and/or splitting points.

Crossing/splitting requirement (area)

If curves cross, you must split the integral at intersection points to avoid cancellation from negative contributions.

Net area vs total area

Net area is ∫ f(x) dx (signed); total area requires splitting at zeros and adding geometric areas (or integrating |f|).

Cross-sectional volume method

A volume technique using V = ∫ A(x) dx (or ∫ A(y) dy), where A is cross-sectional area.

Slice thickness (dx or dy)

The small width of each cross section; multiplying area by thickness gives a small volume element.

Base segment (cross sections)

The segment inside the base region at a slice location whose length determines the cross section’s key dimension.

Square cross sections

Cross sections where area is A(x)=s(x)^2, with side length s(x) determined by the region’s slice length.

Semicircular cross sections

Cross sections shaped as semicircles; if diameter is d, then area A = (1/2)π(d/2)^2.

Diameter-to-radius conversion

A common required step: r = d/2 before using circle or semicircle area formulas.

Perpendicular-to-axis rule (cross sections)

If cross sections are perpendicular to the x-axis, integrate in x; if perpendicular to the y-axis, integrate in y.

Volume of revolution

Volume formed by rotating a plane region about a line, typically computed with disk or washer methods (in this context).

Disk method

Rotation volume with no hole: V = ∫ π(R(x))^2 dx (or ∫ π(R(y))^2 dy).

Washer method

Rotation volume with a hole: V = ∫ π(R^2 − r^2) dx (or dy), using outer radius R and inner radius r.

Outer radius (washer)

The larger distance from the axis of rotation to the outer boundary of the region.

Inner radius (washer)

The smaller distance from the axis of rotation to the inner boundary of the region (creating the hole).

Shifted-axis radius

When rotating about y=k or x=h, the radius is a distance to that line, e.g., R(x)=|f(x)−k| or R(y)=|g(y)−h|.

Perpendicular slicing for disk/washer

For disk/washer, slices must be perpendicular to the axis of rotation (horizontal axis → vertical slices; vertical axis → horizontal slices).

Radius-squaring pitfall

The common mistake in rotation problems of forgetting to square radii in πR^2 or π(R^2−r^2).

Modeling workflow (applications of integration)

A consistent approach: identify the quantity (area/volume/net change/average), choose slices, write the integral with bounds, then check units and reasonableness.