What if the limits of integration are reversed?

Swapping the upper and lower limits flips the sign of the result. Mathematically, $\int_b^a f(x) dx = -\int_a^b f(x) dx$.

Do I need to include the constant of integration $+ C$?

No. When evaluating a definite integral, the constant $C$ is added in $F(b)$ and subtracted in $F(a)$, meaning it will always cancel out to zero.

What happens if the upper and lower limits are the same?

If $a = b$, the width of the interval is zero, so the definite integral evaluates to zero: $\int_a^a f(x) dx = 0$.

How to evaluate a definite integral

A definite integral computes the net signed area under a curve between two specified limits. It is evaluated by finding an antiderivative of the function and applying the second part of the Fundamental Theorem of Calculus. This method applies to any continuous function on a closed interval [a, b].

The setup

Identify the integrand $f(x)$ and the limits of integration, where $a$ is the lower limit and $b$ is the upper limit. Write the integral in the form $\int_a^b f(x) dx$ .

The steps

Find the indefinite integral (antiderivative) $F(x)$ of the integrand $f(x)$ using standard integration rules. Omit the constant of integration $C$ . 2. Evaluate the antiderivative at the upper limit to find $F(b)$ . 3. Evaluate the antiderivative at the lower limit to find $F(a)$ . 4. Subtract the value at the lower limit from the value at the upper limit to compute $F(b) - F(a)$ .

Checking the result

Differentiate your antiderivative $F(x)$ to ensure you recover the original integrand $f(x)$ . Additionally, sketch the function over the interval $[a, b]$ to visually estimate the net area and confirm that the sign and magnitude of your numerical answer make sense.

Common errors

Students frequently mix up the order of subtraction, computing $F(a) - F(b)$ instead of $F(b) - F(a)$ . Another common mistake is dropping negative signs when evaluating terms like $(-x)^2$ or subtracting a negative $F(a)$ value. Finally, forgetting to apply u-substitution properly when finding the antiderivative will yield an incorrect $F(x)$ .

Worked example

Evaluate the definite integral $\int_1^3 (2x + 3) dx$ .

First, find the antiderivative of $f(x) = 2x + 3$ . The antiderivative is $F(x) = x^2 + 3x$ . Next, evaluate $F(x)$ at the upper limit $b = 3$ : $F(3) = (3)^2 + 3(3) = 9 + 9 = 18$ . Then, evaluate $F(x)$ at the lower limit $a = 1$ : $F(1) = (1)^2 + 3(1) = 1 + 3 = 4$ . Finally, subtract $F(a)$ from $F(b)$ : $18 - 4 = 14$ . The definite integral evaluates to 14.

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References: Calculus: Early Transcendentals by James Stewart · OpenStax Calculus Volume 1 · Khan Academy Unit: Definite integrals