Integration and area
Antidifferentiate, then subtract the value at the lower limit from the value at the upper limit.
A worked example
For the integral of (2x + 5) from 1 to 4, an antiderivative is x² + 5x, so the value is (16 + 20) - (1 + 5) = 30.
The mistake people actually make
For the area between two curves, integrate top minus bottom between the x-values where they meet. An indefinite integral needs + C; a definite one does not.