Subtract the lower limit of the integral from the upper limit. Call this h. For example, if you were trying to evaluate y = x^3 between 1 and 3, subtract 3-1 to get 2.
Evaluate the function at the two limits. In the example, x^3 = 27 when x = 3 and 1 when x = 1.
Add these, and multiply by 1/2 h. In the example 27+1 = 28, 28*1/2*2 = 28.
Differentiate the function twice. In the example y' = 3x^2 and y'' = 6x.
Choose various xi (the Greek letter) between the lower and upper limits of the integral, and evaluate the function found in step 4 at the xi. Choose the xi that maximizes the result. (You can try many values using a spreadsheet or many statistical or mathematical programs). In the example, the maximum of 6x between x = 1 and x = 2 occurs at x = 2.
Multiply this maximum by h^3 and divide the result by 12. In the example: 6x at x = 2 = 12. h^3 = 2^3 = 8. 12*8/12 = 8.
Subtract the result in step 6 from the result in step 3 and this is the approximation. In the example 28-8 = 20.