Question

Consider the region below f(x)-(14-x), above the x-axis, and between x = 0 and x = 14 Let xi be the midpoint of the ith subinterval.

Approximate the area of the region using fourteen rectangles. Use the midpoints of each subinterval for the heights of the rectangles.
The area is approximately _____ square units. (Type an integer or decimal.)

Answer 1

To approximate the area of the given region, divide the interval [0, 14] into fourteen subintervals of equal width. Use the midpoints of the subintervals to determine the heights of the rectangles and calculate the area of each rectangle. Add up the areas of all fourteen rectangles to get an approximation of the area of the region.

Step-by-step explanation:

To approximate the area of the region, we will divide the interval [0, 14] into fourteen subintervals of equal width. The width of each subinterval is 14/14 = 1. We will use the midpoints of the subintervals to determine the heights of the rectangles.

For the first subinterval, the midpoint is x₁ = 0.5, and the height is f(0.5) = 14 - 0.5 = 13.5. The area of the first rectangle is 1 * 13.5 = 13.5 square units. Similarly, for the second subinterval, the midpoint is x₂ = 1.5, and the height is f(1.5) = 14 - 1.5 = 12.5. The area of the second rectangle is 1 * 12.5 = 12.5 square units.

We repeat this process for each subinterval and calculate the area of each rectangle. Adding up the areas of all fourteen rectangles will give us an approximation of the area of the region.