Unit 8: Applications of Integration

Integration (as accumulation)

A process that turns a local rate or value into a global total by adding up infinitely many small contributions over an interval.

Riemann sum

A finite sum of sample function values times small widths (e.g., Σ f(x_i)Δx) used to approximate an integral; the integral is the limit as the partition gets finer.

Average value of a function

For a function f on [a,b], the representative “typical” value: favg = (1/(b−a))∫a^b f(x) dx.

Mean value of the function

Another name for the average value of a function on an interval (not the Mean Value Theorem for derivatives).

Interval length (b−a)

The width of the interval [a,b]; it is what you divide by in an average value formula (not b by itself).

Average velocity (via integral)

If velocity is v(t), average velocity on [a,b] is (1/(b−a))∫_a^b v(t) dt.

Rectangle interpretation of average value

If f(x)≥0 on [a,b], f_avg is the constant height of a rectangle with base (b−a) that has the same area as the region under f.

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.

Displacement

The signed change in position over [a,b]: Displacement = ∫_a^b v(t) dt.

Total distance traveled

The total path length traveled over [a,b]: Total distance = ∫_a^b |v(t)| dt.

Fundamental Theorem of Calculus (motion form)

Connects integrals of rates to net change, e.g., s(b)=s(a)+∫a^b v(t)dt and v(b)=v(a)+∫a^b a(t)dt.

Change in velocity

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

Signed area

A definite integral ∫_a^b f(x)dx counts area above the x-axis as positive and area below as negative.

Area between curves (top minus bottom)

If f(x)≥g(x) on [a,b], area between y=f(x) and y=g(x) is A=∫_a^b (f(x)−g(x)) dx.

Intersection points (as bounds)

Points where f(x)=g(x); commonly used as integration limits to capture an enclosed region between curves.

Splitting an integral

Breaking an integral into pieces on subintervals (often at intersection points) to prevent signed-area cancellation when the “top” function changes.

Area using dy (right minus left)

If curves are x=F(y) and x=G(y) with F(y)≥G(y) on [c,d], then area is A=∫_c^d (F(y)−G(y)) dy.

Vertical slice

A thin rectangle of width dx used in area/volume setups; for area between curves it leads to “top minus bottom.”

Horizontal slice

A thin rectangle of height dy used in area/volume setups; for area between curves it leads to “right minus left.”

Volume by cross sections (slicing principle)

If cross-sectional area perpendicular to the x-axis is A(x), then volume is V=∫_a^b A(x) dx (similarly with y).

Cross-sectional area function A(x)

A(x) gives the area of the slice at position x; a thin slice has volume approximately A(x)dx.

Rectangular cross section area

If a cross section is a rectangle with length L(x) and width W(x), then A(x)=L(x)W(x).

Square cross section area

If the slice is a square with side s(x), then A(x)=s(x)^2.

Semicircle cross section area (using diameter)

If the slice is a semicircle with diameter d(x), then A(x)= (π/8) d(x)^2.

Equilateral triangle cross section area

If the slice is an equilateral triangle with side s(x), then A(x)=(√3/4)s(x)^2.

Distance between curves

A common way to get a needed dimension (side length/diameter/height): vertical distance f(x)−g(x) or horizontal distance F(y)−G(y).

Volume of revolution

The volume formed by rotating a 2D region around a line (axis of rotation), computed by integrating volumes of thin circular pieces or shells.

Disk method

Volume method for rotation with no hole: V=π∫_a^b R(x)^2 dx (or with dy), where R is distance to the axis.

Washer method

Volume method for rotation with a hole: V=π∫_a^b (R(x)^2 − r(x)^2) dx, using outer radius R and inner radius r.

Outer radius (R)

In a washer setup, the larger distance from the axis of rotation to the outer boundary of the region.

Inner radius (r)

In a washer setup, the smaller distance from the axis of rotation to the inner boundary (the hole).

Axis of rotation

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

Radius as distance to a shifted axis

When rotating around y=k, a radius is R(x)=|f(x)−k| (typically chosen to be nonnegative without absolute value by using outer/inner curves).

Perpendicular slices (disk/washer criterion)

Disk/washer uses slices perpendicular to the axis of rotation; this guides whether to integrate with dx (vertical slices) or dy (horizontal slices).

Shell method

Volume method using cylindrical shells formed by slices parallel to the axis of rotation; often avoids solving functions for x in terms of y (or vice versa).

Cylindrical shell

A thin hollow cylinder generated by revolving a strip; characterized by radius, height, and thickness.

Shell volume element

For vertical shells: dV = 2π(radius)(height) dx, so V = 2π∫ (radius)(height) dx (similarly with dy for horizontal shells).

Shell radius about x=h or y=k

Distance from the shell to the rotation line, e.g., r(x)=h−x when rotating about x=h (with appropriate orientation).

Shell height

The length of the strip parallel to the axis: typically “top minus bottom” (with y=f(x), y=g(x)) or “right minus left” (with x=F(y), x=G(y)).

Choosing washer vs shell

A strategy decision: both are valid, but choose the method that makes the algebra and bounds simplest (especially avoiding inconvenient rewriting).

Work

Energy transferred by a force acting through a distance; for constant force, W=Fd, and for variable force, it’s computed by an integral.

Work by a variable force

If force along the motion is F(x), work from x=a to x=b is W=∫_a^b F(x) dx.

Units of work (joule)

If force is in newtons and distance in meters, work is in newton-meters, also called joules.

Hooke’s Law

Spring force model: F(x)=kx, where x is displacement from natural length and k is the spring constant.

Spring work integral

Work to stretch/compress a spring from x=a to x=b: W=∫_a^b kx dx.

Pumping/lifting work setup

Uses dW = (force on a slice)(distance lifted); often W=∫ (weight density × area × lift distance) dy.

Weight density (ρ) in pumping problems

A constant converting volume to force (weight): a slice of volume dV has weight dF = ρ dV, leading to integrands like ρA(y)(H−y).

Universal template for applications

Total = ∫ (rate per unit) × (small amount of unit); identify the quantity, choose a variable, write a small piece (dA, dV, dW), then integrate over correct bounds.

Bounds (limits of integration)

The start and end values of the accumulation variable (from given intervals, intersection points, or physical constraints); incorrect bounds can ruin an otherwise correct integrand.

Sign and cancellation pitfall

If you don’t split where needed or ignore absolute values, positive and negative contributions can cancel (e.g., signed area or displacement), giving the wrong “total” quantity.