REFERENCE SHEET

Integration & accumulation

Integrals combine local contributions; the fundamental theorem links accumulation and change.

Before this Limits & continuity, Derivatives & Taylor expansion

\int_{a}^{b} f (x) d x = F (b) - F (a) F^{'} = f

Definitions

Riemann integral

A limit of tagged partition sums as the mesh tends to zero, when all such sums converge to the same number.

Antiderivative

A function F with F′=f on an interval.

Improper integral

A limit of integrals when an endpoint is infinite or the integrand is unbounded; convergence must be checked.

Key theorems

Fundamental theorem of calculus

If f is continuous on [a,b], then A(x)=∫ₐˣ f(t)dt satisfies A′(x)=f(x) in the interior, and any antiderivative evaluates the integral by endpoint subtraction.

Integration by parts

For continuously differentiable u,v on [a,b], ∫ₐᵇ uv′=[uv]ₐᵇ−∫ₐᵇ u′v.

p-integral test

∫₁∞ x⁻ᵖ dx converges exactly for p>1; ∫₀¹ x⁻ᵖ dx converges exactly for p<1.

Notable applications

Probability densities

Integrating a density over a set yields its probability.

Physical accumulation

Integrate velocity for displacement, or force along a line for work.

Work it out

Try before revealing

01 / Compute

Evaluate ∫₀¹ x eˣ dx.

Hint

Take u=x and v′=eˣ.

Solution

An antiderivative is (x−1)eˣ. Evaluation gives 0−(−1)=1.

02 / Decide

Does ∫₁∞ 1/√x dx converge?

Hint

Use p=1/2.