Use two rectangles of equal width to estimate the area between the graph of f(x) = x - cos(πx) and the x-axis on the interval [1, 5]. Evaluate the function at the mid-point of each rectangle to find each height.

Question

a) 8
b) 12
c) 16
d) 20

thank you in advance.

Answer 1

use two rectangles on [1,5] means they both have width 2.
So, the 2 subintervals are [1,3] and [3,5]
The midpoints are thus at x=2,4
So, take f(2) and f(4) for the heights.
Your approximation is just the sum of those two rectangles.

Answer 2

Thanks but can you explain me more because I am so so so confuse

Answer 3

what? You cannot evaluate f(x) at x=2 and x=4?
The height of the rectangles is
f(2) = 2 - cos(2π) = 2-1 = 1
f(4) = 4 - cos(4π) = 4-1 = 3
Since the width of each rectangle is 2, the area
a = 2*1 + 2*3 = 8

Now study your text's examples again, and it should make more sense.
If not, consider asking your teacher (or a tutor) for some one-on-one help.
youtube also has videos explaining quadrature.