Definite Integrals

For a continuous function f(x) on

For a continuous function f(x) on [a,b], the definite integral ∫[a to b]f(x)dx = lim(n→∞) (1/n)[f(a) + f(a+h) + ... + f(a+(n-1)h)] where h = (b-a)/n.

If F(x) is an antiderivative

If F(x) is an antiderivative of f(x), then ∫[a to b]f(x)dx = F(b) - F(a) = [F(x)]ᵇₐ. Step-by-step method: 1) Find antiderivative F(x) of f(x), 2) Eval

Seven key properties that simplify evaluation

Seven key properties that simplify evaluation: 1) ∫[a to b]f(x)dx = -∫[b to a]f(x)dx (order reversal), 2) ∫[a to c]f(x)dx = ∫[a to b]f(x)dx + ∫[b to c

When using substitution u = g(x)

When using substitution u = g(x), remember to change limits: if x = a gives u = α and x = b gives u = β, then ∫[a to b]f(g(x))g'(x)dx = ∫[α to β]f(u)d

Three main types

Three main types: 1) Area under curve y = f(x) from x = a to x = b: A = ∫[a to b]f(x)dx, 2) Area between two curves y = f(x) and y = g(x) where f(x) >