. Compute the left and right Riemann sums—L4 and R4, respectively—for f(x)=(2−|x|) on [−2,2]. Compute their average value and compare it with the area under the graph of f.

Question

please try to do it step by step.

Answer 1

n = 4

Δx = 2-(-2)/4 = 4/4 = 1

L4: Δx∑f(x_i-1) = (1)[0+1+2+1] = 4

R4:Δx∑f(x_i) = (1)[1+2+1+0] = 4

Average = (4+4)/2 = 4

The area under the curve is the area of a triangle with base of 4 and height of 2.

A_triangle = base•height/2 = 4•2/2 = 4

All four values compare.

Answer 2

n = 4;Δx = (2 - (-2))/4 = 1

L₄ = (0 + 1 + 2 + 1)·1 = 4

R₄ = (1 + 2 + 1 + 0)·1 = 4

(L₄ + R₄)/2 = 4

Area A = 4·2/2 = 4