Set up the integral if the problem does not give it to you. If you need to find the area under the curve 3x^2 - 2x + 1 between 1 and 3, for example, you need to take the integral of the function over that interval: int[(3x^2 - 2x + 1)dx] from 1 to 3.
Use the basic rules of integration to solve the integral the same way you would for an indefinite integral, but do not add the constant of integration. For example, int[(3x^2 - 2x + 1)dx] = x^3 - x^2 + x.
Substitute the upper limit of the interval of integration for x in the resulting equation and simplify. For example, replacing x with 3 in x^3 - x^2 + x results in 3^3 - 3^2 + 3 = 27 - 9 + 3 = 21.
Replace x with the lower limit of the interval in the result of the integral and simplify. For example, substituting one into x^3 - x^2 + x gives 1^3 - 1^2 + 1 = 1.
Subtract the lower limit from the upper limit to get the result of the definite integral. For example, 21 - 1 = 20.