Numerical Integration Example using Parabolas (Simpson’s Rule)

Estimate the area under the curve y(x) = x² from x = 1 to x = 3 .

Note: This function happens to be an "increasing" function. (concave up)

Here Δx = (b – a)/3n Restriction: You need an even number of segments!

Note: The approximate area = ( y0 + 4y1 + 2y2 + 4y3 + y4 ) Δx

Here b = 3, a = 1, n = 4 so Δx = (3 – 1)/12 = 0.1666666

y0 = 1, y1 = 2.25, y2 = 4, y3= 6.25, and y4 = 9

Approximate area = ( 1.0 + 4x2.25 + 2x4 + 4x6.25 + 9 ) 0.16666 = 8.66666

Note: Since the function was a parabola, the approximation using parabolic

segments yields the exact area, 26/3 = 8.66666.

Click here to return to the beginning of numerical integration.

Return to Notes for Calculus 1

Copyright © 2019 Richard C. Coddington

All rights reserved.