Can you find the answer to the area of the region and exact area that is obtained?

Question

Use the rectangles to approximate the area of the region. f(x) = -x + 11 [1,11] answer:_____

Give the exact area obtained using a definite integral.

answer:____

Answer 1

There are 4 rectangles, each width 2.5:

They are evaluated at the midpoint:

Now, integrate (-x+11) from 1 to 11.

Answer 2

Approximate area is:

(3,75-1)*8.75 + (6-3.75)*6.25 + (8.25-6)*4 + (11-8.25)*1 = 49.875

the integral is (-1/2)x^2 + 11x

x=11 ---> (-121/2) + 121 = 121 - 121/2 = 121/2

x=1 ---> (-1/2) + 11 = 10.5

121/2 - 21/2 = 100/2 = 50