Numerical Integration Example using the Center point (also called midpoint)

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)

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

Further Note: In this application use of midpoint sampling will provide

an improved estimation of the actual area.

Here b = 3, a = 1, n = 4 so Δx = (3 – 1)/4 = 0.5

c1 = 1.25, c2 = 1.75, c3 = 2.25, and c4= 2.75 note here yi = ci²

y1 = 1.5625, y2 = 3.0625, y3 = 5.0625, y4 = 7.5625

Approximate area = ( 1.25 + 1.75 + 2.25 + 2.75 ) 0.5 = 8.625

Exact value of area = 26/3 = 8.66666

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