The art of accumulation — find areas, volumes, and totals by summing infinitely many infinitesimal pieces.
An antiderivative (or indefinite integral) of f(x) is any function F(x) whose derivative is f(x):
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More antiderivatives can be found on the formula sheet.
The definite integral computes the signed area between f(x) and the x-axis from a to b:
This is a limit of Riemann sums — rectangles approximating the area. The connection to probability is direct: the probability of a continuous random variable falling in [a,b] is exactly ∫ₐᵇ f(x) dx where f is the probability density function.
The FTC links differentiation and integration — two seemingly opposite operations are inverses:
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The integral version of the chain rule. Let u = g(x), du = g'(x)dx:
Let u = x², du = 2x dx
∫ cos(u) du = sin(u) + C = sin(x²) + C
The integral version of the product rule:
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= xeˣ − ∫ eˣ dx = xeˣ − eˣ + C
Decompose a rational function into simpler fractions. Requires factoring the denominator first:
Use trig identities to handle expressions involving √(a² − x²), √(a² + x²), or √(x² − a²).
These connect integration to 3D geometric shapes — computing volumes that geometry formulas alone can't handle.
For a continuous probability distribution with density f(x):
The normal distribution, exponential distribution, and every other continuous distribution is defined through integrals.