Use the given graph to estimate the left Riemann sum for the

Use the given graph to estimate the left Riemann sum for the given interval with the stated number of subdivisions. HINT [See Example 3.]

[1, 3], n = 4

Solution

From the data given above, lower limit is x = 1 and upper limit is x = 3

our job is to fit 4 [n = 4] rectangles within this region and find the sum of the area of these rectangles. This will be a Reimann sum [approximation] for the given integral.

d = (3 - 1)/4 = 1/2

so the width of all the 4 rectangles is d = 4

A = [4 x f(1.25)] + [4 x f(1.75)] + [4 x f(2.25)] + [4 x f(2.75)]

here, the value of the function is chosen at the midpoint of the two values so as to take an average approximation of the value of the function at both the upper and the lower limit of that rectangle.

so A = [4 x 3.4] + [4 x 3.4] + [4 x 1] + [4 x 1] = 35.20 sq units.