Unit 8: Applications of Integration

Average value of a function

The continuous version of an average over an interval, found by integrating the function over the interval and dividing by the interval length.

Average value formula

For f on [a,b], the average value is favg = (1/(b−a))∫a^b f(x) dx.

Interval length (b−a)

The width of the interval [a,b]; the quantity you divide by when computing average value.

Definite integral

An accumulation (limit of sums) that measures net change or signed area over an interval, written ∫_a^b f(x) dx.

Signed area

Area counted positive when a graph is above the x-axis and negative when it is below; this is what a definite integral represents geometrically.

“Typical height” interpretation of average value

The constant height that would produce the same signed area over [a,b] as the original function.

Mean Value Theorem for Integrals

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.

Average value vs. average of endpoints

In general, f_avg is not equal to (f(a)+f(b))/2; that endpoint average is tied to trapezoidal rule ideas, not the exact average value.

Trapezoidal rule (connection)

A numerical approximation method where (f(a)+f(b))/2 appears; it does not usually give the exact average value of a function.

Average value vs. average rate of change

Average value averages outputs of f; average rate of change is (f(b)−f(a))/(b−a) and measures a slope across the interval.

Displacement

Net change in position over a time interval; computed by integrating velocity: ∫ v(t) dt.

Total distance traveled

Total path length traveled; computed by integrating speed, i.e., ∫ |v(t)| dt.

Velocity

The derivative of position; a rate of change of position with respect to time (units: distance/time).

Acceleration

The derivative of velocity; a rate of change of velocity with respect to time (units: distance/time^2).

Fundamental Theorem of Calculus (net change idea)

A definite integral of a rate of change over [a,b] gives the net change of the original quantity over that interval.

Acceleration-to-velocity relationship (FTC form)

∫_a^b a(t) dt = v(b) − v(a), so the integral of acceleration gives change in velocity.

Net change

Final value minus initial value; what you get from integrating a rate function over an interval.

Speed

The magnitude of velocity, |v(t)|; integrating it gives total distance traveled.

Area between curves

The (nonnegative) geometric area enclosed between two graphs, computed by integrating a distance between them.

Top minus bottom (vertical slices)

For area with respect to x, use ∫_a^b (top function − bottom function) dx when the top curve stays above the bottom curve.

Right minus left (horizontal slices)

For area with respect to y, use ∫_c^d (right boundary − left boundary) dy after rewriting curves as x = R(y) and x = L(y).

Intersection points (for bounds)

Points where curves meet, found by solving f(x)=g(x); their x-values often determine integration limits for enclosed regions.

Splitting an integral

Breaking an integral into pieces when the integrand rule changes (e.g., when the top curve switches), to avoid cancellation and get correct geometric area.

Cancellation (in signed area)

When positive and negative contributions in a single integral offset each other; this is why you must split integrals for geometric area if curves cross.

Test point method

A quick check (plug in a value) used to determine which function is on top/bottom (or right/left) on a subinterval.

Vertical slices

Using rectangles/slabs perpendicular to the x-axis; typically leads to integrals in dx.

Horizontal slices

Using rectangles/slabs perpendicular to the y-axis; typically leads to integrals in dy.

Cross section

The 2D shape obtained by slicing a 3D solid; its area A(x) or A(y) is integrated to find volume.

Slicing method for volume

Volume is found by integrating cross-sectional area: V = ∫a^b A(x) dx (or V = ∫c^d A(y) dy).

Riemann sum volume model

An approximation for volume using thin slices: volume ≈ Σ A(x_i)Δx; the integral is the limit as Δx→0.

Slice thickness (Δx or Δy)

The small width of each slab in a volume approximation; becomes dx or dy in the integral.

Base region (for cross sections)

The 2D region in the xy-plane that determines the width of each slice used to build cross-sectional area.

Width of a slice (vertical)

For cross sections based on vertical slices, width w(x) is typically top(x) − bottom(x) from the base region.

Width of a slice (horizontal)

For cross sections based on horizontal slices, width w(y) is typically right(y) − left(y) from the base region.

Square cross section area (in terms of width)

If the cross section is a square with side w, then A = w^2.

Rectangle cross section with proportional height

If height = k·w for constant k, then A = w(k·w) = k w^2.

Semicircle cross section area (diameter w)

If the diameter is w, then radius is w/2 and A = (1/2)π(w/2)^2 = (π/8)w^2.

Circle cross section area (diameter w)

If the diameter is w, then radius is w/2 and A = π(w/2)^2 = (π/4)w^2.

Equilateral triangle cross section area (side w)

If the side length is w, then A = (√3/4)w^2.

Even function symmetry (integration shortcut)

If f is even, then ∫{−a}^a f(x) dx = 2∫0^a f(x) dx; useful in symmetric volume setups.

Volume of revolution

The 3D volume formed by rotating a 2D region around a line (axis of rotation), computed with integrals.

Axis of rotation

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

Disk method

A volume method for rotation that uses solid circular cross sections: V = π∫_a^b (R(x))^2 dx (no hole).

Washer method

A volume method for rotation that uses circular cross sections with a hole: V = π∫_a^b (R(x)^2 − r(x)^2) dx.

Outer radius (R)

The larger distance from the axis of rotation to the outer boundary of the region; used in washers as R^2.

Inner radius (r)

The smaller distance from the axis of rotation to the inner boundary of the region (the hole); used in washers as r^2.

Radius as “distance to the axis”

In revolution problems, a radius is a perpendicular distance from the axis of rotation to a curve, often written |f(x)−k| or |x−h|.

Choosing dx vs. dy (revolution)

Use dx with vertical slices (often when y=f(x) and rotating about a horizontal line); use dy with horizontal slices (often when x=g(y) and rotating about a vertical line).

Modeling workflow for integral setup

Sketch, choose slicing direction, write the key distance/area expression, pick correct bounds, build the integral with dx or dy, then evaluate if needed.

Sanity check: nonnegative area/volume

Geometric area and volume should be nonnegative; a negative result usually signals reversed subtraction or a missing split.