Area/accumulation viewpoint (definite integrals)

Antiderivative viewpoint (indefinite integrals)

Distance from small distance increments

Mass from small mass elements

Total cost from small cost contributions

Rate (per unit time) → total over time

Density (per unit length/area/volume) → total over a region

∫_a^b f(x) dx

a, b: start and end of accumulation

f(x): accumulation rate/density at position x

dx: an infinitesimal “width” element indicating the variable of accumulation

Split [a, b] into many small subintervals

Approximate total by summing rectangles

Take the limit as Δx → 0 to get the exact accumulated total

If f(x) ≥ 0 on [a, b], the integral equals area under the curve

If f(x) dips below 0, it subtracts (signed area)

Define F(x) = ∫_a^x f(t) dt

F(x) tracks how much has accumulated up to x

F'(x) = f(x) (rate of accumulation equals the integrand)

∫ f(x) dx = F(x) + C

Find a function whose derivative is f(x)

Represents a whole family of functions differing by a constant

Antiderivatives provide a shortcut to compute definite integrals via evaluation

If F(x) = ∫_a^x f(t) dt, then F'(x) = f(x) (under mild conditions)

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

Converts “sum infinitely many pieces” into “evaluate an antiderivative at endpoints”

If v(t) is velocity, displacement over [t1, t2] is ∫_{t1}^{t2} v(t) dt

If v(t) can be negative, integral yields net displacement (direction included)

Total distance traveled often requires ∫ |v(t)| dt

If r(t) is a growth/decay rate of quantity Q, then change in Q is ∫ r(t) dt

Line density λ(x): mass on [a, b] is ∫_a^b λ(x) dx

Area density ρ(x, y): mass over region R is ∬_R ρ(x, y) dA

If C'(q) is marginal cost, total additional cost from q=a to q=b is ∫_a^b C'(q) dq

Example: (meters/second) × (seconds) = meters

Example: (kg/m) × (m) = kg

Double integrals ∬_R f(x, y) dA accumulate over area

Triple integrals ∬∬_V f(x, y, z) dV accumulate over volume

Line integrals accumulate along paths (e.g., work along a curve)

Surface integrals accumulate across surfaces (e.g., flux)

∫_a^b f(x) dx = ∫_a^c f(x) dx + ∫_c^b f(x) dx

∫ (af + bg) = a∫ f + b∫ g

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

Use antiderivatives and FTC when possible

Techniques

Riemann sums (left/right/midpoint)

Trapezoidal rule

Simpson’s rule

Useful when no elementary antiderivative exists or data is discrete

It equals signed area; negative values subtract

It indicates the variable and the infinitesimal element being accumulated

Indefinite: family of functions; definite: a number (net accumulated total)