Subtract the lower limit of the integral from the upper. Divide the result by four. For example, if you want to find the area under the curve y = x^8 from 0 to 1. Subtract 1 - 0 = 1, divide by 4 = 2.5 and the intervals are 0-.25, .25-.5, .5-.75 and .75-1.0. Call these f1, f2, f3, f4 and f5. Call the size of the interval h.
Differentiate the function 6 times. In the example, the first derivative of x^8 is 8x^7, the second derivative is 56x^6, the third is 336x^5, the fourth is 1,680x^4, the fifth is 6,720x^3 and the sixth is 20,160x^2.
Find the point on the whole interval where this function is maximized. Evaluate the function at that point. In the example, the maximum for 20,160x^2 between 0 and 1 is at 1, where it equals 20,160.
Raise h to the 7th power. In the example, .25^7 = .000061.
Multiply the result from step four by 8/945, then multiply by the results in step three. In the example, .0000610 x 8/945 x 2,016 = 0.001042.
Calculate (7 x f1) + (32 x f2) + (12 x f3) + (32 x f4) + (7 x f5). In the example this is 0 + 8 + 6 + 24 + 7 = 45.
Multiply the result from step six by 2/45 multiplied by h. In the example, 45 x 2/45 x .25 = .5.
Subtract the result in step five from the result in step seven to find the estimate. In the example, .5 - .0010402 = 4.998958.