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Calculus · Quick Reference
Definite Integrals Cheat Sheet
In one line: a definite integral computes a specific numeric value — often representing the net area between a curve and the x-axis over a given interval.
Key Ideas
1Definite Integral Notation. Integral from a to b of f(x) dx represents the net signed area under f(x) between x=a and x=b.
2Fundamental Theorem of Calculus. If F is an antiderivative of f, then Integral from a to b of f(x) dx = F(b) - F(a).
3Signed Area. Area above the x-axis counts as positive; area below counts as negative.
4Properties of Definite Integrals. Integral from a to a = 0. Integral from a to b = -Integral from b to a. Integrals over adjoining intervals add together.
5No '+C' Needed. Since we evaluate F(b) - F(a), any constant C cancels out, so definite integrals give one exact number.
Worked Examples
Evaluate the integral from 0 to 2 of x^2 dx.
8/3
Evaluate the integral from 1 to 3 of (2x + 1) dx.
10
Evaluate the integral from -1 to 1 of x^3 dx.
0 (the positive and negative areas cancel out)
Formulas
Integral from a to b of f(x) dx = F(b) - F(a)
Practice Yourself
Evaluate the integral from 0 to 1 of x^2 dx.
1/3
Evaluate the integral from 0 to 3 of 2 dx.
6
Evaluate the integral from 1 to 2 of x dx.
3/2