Use the rectangles to approximate the area of the region. Give the exact area obtained using a definite integral.

Question

f(x) = 8

x

      [1, 5]

Answer 1

y = 8 is a straight line. You are solving for the area of a rectangle that is 4 units long in x direction and 8 units high in the y direction. It's just 4 * 8 = 32.

x direction - length is | 1-5| = |-4| = 4.

y is the line y = 8.

The integral is

∫₁⁵ 8 dx

= [8y]₁⁵

= 8×5 - 8×1 = 40 - 8 = 32.

I'm not sure if you wanted the integral as well, but there it is.

Answer 2

integral is 8x evaluated between 1 and 5 = 40-8 = 32. It's too obvious, f(x) = 8 is a flat horizontal line, creating a rectangle of 8 x 4 = 32

use as many rectangles as you want, one billion rectangles you'll get nearly exactly 32