he 5 rectangles in the graph below illustrate a midpoint Riemann sum for f(x)=2^x on the interval [1,4]. The value of this Riemann sum is .

Question

Answer 1

There are 5 rectangles in this Riemann sum. Assuming each rectangle has the same width, that width is (4-1)/5 = 3/5.

To determine the x_i midpoints to plug into the function to get the height of each rectangle:

x₁= 1 + (1/2)(3/5) [start at 1 and add on half of a width] = 1.3

x₂=1+(3/2)(3/5) [start at 1 and add on 1 and 1/2 widths] = 1.9

x₃ = 1 + (5/2)(3/5) = 2.5

x₄=1+(7/2)(3/5) = 3.1

x₅=1+(9/2)(3/5)=3.7

The height of each rectangle is found by plugging those values into the function. And the area is found by multiplying each of those heights by the width (3/5). Since each area has a factor of 3/5, consider factoring that value out as the areas are added together:

A = (3/5) [ 2^1.3 + 2^1.9 + 2^2.5 + 2^3.1 + 2^3.7]