How to Calculate an Area Using Simpson's Rule

The solution to a definite integral represents the area under the curve formed by the equation between the upper and lower limits of the integral. Some equations, however, are complicated to integrate. Simpson's rule provides one method of approximating the area under the curve for such equations. Dividing the area under the curve with several vertical lines, connecting each set of three lines with a parabola, and summing the areas under the parabolic curves will give you an approximation of the total area under the curve.

Things You'll Need

  • Calculator

Instructions

    • 1

      Divide the area under the curve into an even number of equally spaced intervals along the x axis. If you want to find the area under the curve from 0 to 8, for example, you might divide the area into four intervals, each having a width of two in the x direction.

    • 2

      Subtract the lower limit of x from the higher limit of x and divide the result by the number of intervals. For an area from 0 to 8 divided into four intervals, use the equation (8 - 0)/4 = 2.

    • 3

      Divide the result by three -- in this example, you would get 2/3. Record this number for later use.

    • 4

      Calculate the value of f(x) at each division along the curve, starting with the lower limit and ending with the upper limit. For a curve from 0 to 8 divided into four intervals, calculate the values of f(x) at 0, 2, 4, 6 and 8. If the equation of the curve is f(x) = x^2 - x + 2, for example, calculate f(0) = 2, f(2) = 4, f(4) = 14, f(6) = 32 and f(8) = 58.

    • 5

      Multiply the second value of f(x) by 4. Multiply the third value of f(x) by 2. Continue multiplying with this pattern until you reach the next-to-last value of f(x), which should be multiplied by 4. Add all the values together; for example, f(0) + 4f(2) + 2f(4) + 4f(6) + f(8) = 2 + 44 + 214 + 432 + 58 = 168.

    • 6

      Multiply the result of step 5 by the result of step 3. For example, (2/3)*168 = 112. This result approximates the area under the curve.

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