Use the Midpoint Rule with to approximate the area of the region bounded by the graph of the function and the x-axis over the interval.Sketch the region.

Question

Use the Midpoint Rule with  to approximate the area of the region bounded by the graph of the function and the x-axis over the interval.Sketch the region. ​  ​

Quizplus · Accepted Answer

Using the midpoint rule, we divide the interval [0,5] into n=4 subintervals of equal width:
Δx = (5-0)/4 = 1.25
Then, we evaluate the function at the midpoint of each subinterval and multiply by the width:
f(0.625)Δx + f(1.875)Δx + f(3.125)Δx + f(4.375)Δx
= 4(-0.5)1.25 + 4(2.5)1.25 + 4(-2.5)1.25 + 4(0.5)1.25
= -2.5 + 12.5 - 12.5 + 2.5
= 0
Therefore, the approximate area is zero, which means the best choice is C since it is the closest to zero. We can also verify this by looking at the graph: the region is symmetric around the x-axis and has positive and negative areas that cancel out.

Use the Midpoint Rule with to Approximate the Area