Definite Integral: Step-by-Step Guide
A definite integral computes the net signed area between a function and the x-axis over a closed interval [a, b]. By the Fundamental Theorem of Calculus, if F is any antiderivative of f, then the integral from a to b of f(x) dx equals F(b) - F(a).
Math Racer skill guide · Calculus 2 · AP Calc BC / College
How to solve it, step by step
- Identify the integrand and limits. Read off f(x), the lower limit a, and the upper limit b. Make sure f is continuous (or at least integrable) on [a, b].\[\int_{a}^{b} f(x)\,dx\]
- Find an antiderivative. Compute F(x) so that F'(x) = f(x). Use power rule, trig identities, exponentials, or substitution as needed. You may drop the +C since it cancels.\[F(x) = \int f(x)\,dx\]
- Apply the Fundamental Theorem of Calculus. Evaluate F at the upper limit and subtract its value at the lower limit.\[\int_{a}^{b} f(x)\,dx = F(b) - F(a)\]
- Simplify and check. Combine fractions, simplify radicals, and verify that the sign and magnitude make sense given the graph of f on [a, b].\[F(b) - F(a) = \text{numerical value}\]
Worked example
Evaluate the integral from 1 to 3 of (2x + 1) dx.
Step 1: Identify f(x) = 2x + 1, a = 1, b = 3.
Step 2: Find an antiderivative F(x) = x^2 + x.
Step 3: Evaluate at the bounds: F(3) = 9 + 3 = 12 and F(1) = 1 + 1 = 2.
Step 4: Subtract: F(3) - F(1) = 12 - 2 = 10.
Step 5: The definite integral equals 10.
Verification: The integrand 2x + 1 is linear, so the region from x=1 to x=3 is a trapezoid with parallel sides 3 and 7 and width 2. Area = (1/2)(3+7)(2) = 10. Matches.
Tips and common mistakes
- Drop the +C constant when computing definite integrals; it cancels in F(b) - F(a).
- If you change variables via u-substitution, change the limits of integration to u-values to avoid back-substituting.
- Watch for sign issues: integral from a to b equals minus integral from b to a.
- Split the interval at any discontinuities or where the integrand changes sign.
- For symmetric intervals, exploit even/odd symmetry: integral of an odd function over [-a, a] is 0.
- Sketch the curve to predict whether the result should be positive, negative, or zero.
- Always simplify the antiderivative before plugging in the bounds; messy F(x) leads to arithmetic errors.