An integral adds up many tiny contributions to measure a total

Often described as continuous summation

Area under a curve

Total change from a rate

Mass from density

Work from force along a distance

Probability from a density function

Break a quantity into tiny intervals or elements

Approximate each piece, then add them

Make pieces smaller to improve accuracy

In the limit, the approximation becomes exact (under suitable conditions)

Choose an interval [a, b]

Split into subintervals: a = x0 < x1 < … < xn = b

Widths: Δxi = xi − x(i−1)

Pick a point ci in each subinterval [x(i−1), xi]

Evaluate height: f(ci)

Sum of height × width: Σ f(ci) Δxi

Interpretation: total of many small contributions

Make the largest subinterval width go to 0

If the limit exists, it defines the definite integral

∫a^b f(x) dx

dx indicates accumulation with respect to x (the tiny width pieces)

Result is a number representing the total accumulated amount on [a, b]

Above x-axis contributes positively

Below x-axis contributes negatively

Net accumulation = positive contributions − negative contributions

If f(x) has units U and x has units X

Then ∫ f(x) dx has units U·X (accumulated total)

∫ f(x) dx = F(x) + C where F′(x) = f(x)

Represents a family of functions whose derivative is f

Define A(x) = ∫a^x f(t) dt

A(x) measures accumulated total from a to x

A′(x) = f(x) (the accumulation rate equals the integrand)

If A(x) = ∫a^x f(t) dt, then A′(x) = f(x)

Interpretation: the instantaneous rate of growth of the accumulated total is f

If F′(x) = f(x), then ∫a^b f(x) dx = F(b) − F(a)

Interpretation: total accumulation equals net change in an antiderivative

If v(t) is velocity, displacement = ∫t1^t2 v(t) dt

If v(t) ≥ 0, distance traveled = ∫ v(t) dt

Area under y = f(x) from a to b is ∫a^b f(x) dx (when f ≥ 0)

1D rod with linear density ρ(x): mass = ∫a^b ρ(x) dx

2D lamina with density ρ(x, y): mass = ∬ ρ(x, y) dA

Constant direction: W = ∫a^b F(x) dx

Along a path: W = ∫ F · dr (line integral)

If X has pdf p(x): P(a ≤ X ≤ b) = ∫a^b p(x) dx

Total probability: ∫−∞^∞ p(x) dx = 1

Represents the width (or measure) of each small piece in x

Adds infinitely many infinitesimal contributions

∫ f(x) dx accumulates with respect to x

Changing variable changes what small pieces mean (e.g., dt, dθ)

∫ (a f(x) + b g(x)) dx = a ∫ f(x) dx + b ∫ g(x) dx

∫a^b f(x) dx = ∫a^c f(x) dx + ∫c^b f(x) dx

∫b^a f(x) dx = −∫a^b f(x) dx

If f(x) ≥ g(x) on [a, b], then ∫ f ≥ ∫ g

If m ≤ f(x) ≤ M, then m(b−a) ≤ ∫a^b f ≤ M(b−a)

Continuous functions on [a, b] are integrable

Functions with only finitely many jump discontinuities are integrable

Highly irregular functions may fail Riemann integrability

Infinite intervals or unbounded functions require improper integrals

∫a^∞ f(x) dx defined as lim(b→∞) ∫a^b f(x) dx

If f(x) blows up at c, define via limits around c

Convergent: finite accumulated total

Divergent: total grows without bound or oscillates without settling

∬R f(x, y) dA accumulates over a region in the plane

Example: volume under a surface over region R

∭V f(x, y, z) dV accumulates over a 3D region

Example: mass from volumetric density

Accumulate along curves (line) or across surfaces (surface)

Used for work, flux, circulation, field totals