Question

Evaluate the upper and lower sums for f(x) = 1 x², -1 ≤ x ≤ 1 with n = 3 and 4. Unless specified, all approximating rectangles are assumed to have the same width.

Answer 1

To evaluate the upper and lower sums for f(x) = x², -1 ≤ x ≤ 1 with n = 3 and 4, divide the interval into subintervals of equal width. The lower sum is the sum of the areas of the rectangles formed by the lower function values over each subinterval. The upper sum is the sum of the areas of the rectangles formed by the upper function values over each subinterval.

Step-by-step explanation:

To evaluate the upper and lower sums for the function f(x) = x², -1 ≤ x ≤ 1 with n = 3 and 4, we divide the interval [-1, 1] into n subintervals of equal width.

For n = 3, the width of each subinterval is Δx = (1 - (-1)) / 3 = 2/3.

The lower sum is the sum of the areas of the rectangles formed by the lower function values over each subinterval. For n = 3, the lower sum is (1/9) + (4/9) + (1/9) = 6/9.

The upper sum is the sum of the areas of the rectangles formed by the upper function values over each subinterval. For n = 3, the upper sum is (4/9) + (1) + (4/9) = 26/9.

For n = 4, the width of each subinterval is Δx = (1 - (-1)) / 4 = 1/2.

The lower sum is (1/16) + (1/4) + (1/4) + (1/16) = 9/16.

The upper sum is (1/4) + (1) + (1) + (1/4) = 17/4.