Definite integrals

The essentials

The definite integral ∫ from a to b of f(x) dx can be seen as the limit of a sum: divide [a, b] into many thin strips and add up f(x) × width. For a continuous non-negative f, it equals the area under the curve between x = a and x = b.

The second fundamental theorem of calculus makes evaluation easy: if F is any antiderivative of f, then ∫ₐᵇ f(x) dx = F(b) − F(a). The constant C cancels, so it is not written. The first fundamental theorem says that the area function A(x) = ∫ₐˣ f(t) dt has derivative f(x).

Key points

• ∫ from a to b of f(x) dx = −(∫ from b to a of f(x) dx), and ∫ₐᵃ f(x) dx = 0.
• ∫ from a to b = (∫ from a to c) + (∫ from c to b), for any c between a and b.
• ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a − x) dx (very useful with sin and cos on [0, π/2]).
• ∫₋ₐᵃ f(x) dx = 2∫₀ᵃ f(x) dx if f is even, and 0 if f is odd.
• In substitution for a definite integral, change the limits as well.
• Example: ∫₀^(π/2) sin x/(sin x + cos x) dx: call it I; replacing x by π/2 − x gives the same integral with cos on top, and adding the two forms gives 2I = π/2, so I = π/4.

Exam tips

• Before integrating, check whether the integrand is odd on a symmetric interval; the answer may simply be 0.
• When an integral looks hard, try the a − x property and add the two versions.

Common mistakes

• Subtracting in the wrong order: it is F(upper) − F(lower).
• Keeping the old limits after a substitution.