Question

(a) Estimate the area under the graph of f(x) = 2 cos(x) from x = 0 to x = π/2 using four approximating rectangles and right endpoints. (Round your answers to four decimal places.)
(b) Repeat part (a) using left endpoints

Answer 1

  first, we are going to define variables as the following: 

a = 0 
a = π/2 
n = 4 rectangles 

Δx = [ b - a ] / n ------>Δx = [ π/2 - 0 ] / 4 = π/8 

right endpoints : 
sum( seq( 4 cos(x) * π/8 , x , 0+π/8 , π/2 , π/8 ) ) = 3.163065 underestimate 

left endpoints: 
sum( seq( 4 cos(x) * π/8 , x , 0 , π/2-(π/8) , π/8 ) ) = 4.733861 overestimate 

the reason because the actual estimate by integral as the following: 
π/2 
∫ 4cos(x) dx = 4 
0