If f(x) = (e^x) − 7, 0 ≤ x ≤ 2, find the Riemann sum with n = 4 correct to six decimal places, taking the sample points to be midpoints

Question

If f(x) = e^x − 7, 0 ≤ x ≤ 2, find the Riemann sum with n = 4 correct to six decimal places, taking the sample points to be midpoints

Answer 1

With n = 4 and the interval x ∈ [0 , 2] then the subintervals are [0 , .5] , [.5 , 1] , [1 , 1.5] , and [1.5 , 2].

The midpoints of those subints (all of width = .5) are x = .25 , .75 , 1.25 , and 1.75

The heights at those x-values are e^.25 - 7 , e^.75 - 7 , e^1.25 - 7 , and e^1.75 - 7.

So the sum of the areas of those 4 rectangles = .5(e^.25 - 7 + e^.75 - 7 + e^1.25 - 7 + e^1.75 - 7).