Add small contributions across an interval
A definite integral accumulates contributions across a range. In a graph interpretation it gives signed area: regions below the horizontal axis contribute negatively.
Use an antiderivative when the conditions apply
For a continuous integrand on the interval, an antiderivative lets you evaluate the definite integral by subtracting its value at the lower bound from its value at the upper bound. Check that differentiating the antiderivative returns the integrand.
A mistake to avoid
An indefinite integral describes a family of antiderivatives and includes a constant. A definite integral uses bounds and evaluates to a value; adding an unexplained constant to that result changes the answer.
Worked example
Evaluate the integral of 2x from 0 to 3
An antiderivative of 2x is x². Evaluate at the bounds: 3² − 0² = 9. The graph is nonnegative on this interval, so the same value is the area of a triangle with base 3 and height 6: (3 × 6)/2 = 9.