Estimate the area above the x-axis and under the graph of f(x)=4x^3+6 from x=-1 to x=5, first using 6 approximating rectangles and right endpoints

Question

(A) Estimate the area above the x-axis and under the graph of f(x)=4x^3+6

from x=-1 to x=5, first using 6 approximating rectangles and right endpoints, and then improving your estimate using 12 approximating rectangles and right endpoints.



Area of 6 Rectangles = 

Area of 12 Rectangles = 

(B) Repeat part (A) using left endpoints.

Area of 6 Rectangles = 

Area of 12 Rectangles = 

(C) Repeat part (A) using midpoints.

Area of 6 Rectangles = 

Area of 12 Rectangles 

Answer 1

Area of 6 rectangles = ∑_i=1,6f(x₁+(i-1)*Δx)*Δx, where Δx=(x_end-x_start)/6

Area of 12 rectangles = ∑_i=1,12f(x₁+(i-1)*Δx)*Δx, where Δx=(x_end-x_start)/12

A) For 6 rectangles and right endpoints x₁=0, Δx=1

Area of 6 rectangles = (f(0)+f(1)+f(2)+f(3)+f(4)+f(5))*1 = 6+10+38+114+262+506 = 936

For 12 rectangles and right endpoints x₁=-0.5, Δx=0.5

Area of 12 rectangles = (f(-0.5)+f(0)+f(0.5)+f(1)+f(1.5)+f(2)+f(2.5)+f(3)+f(3.5)+f(4)+f(4.5)+f(5))*0.5

= (5.5+6+6.5+10+19.5+38+68.5+114+177.5+262+370.5+506)*0.5 = 792

B) For 6 rectangles and left endpoints x₁=-1, Δx=1

Area of 6 rectangles = (f(-1)+f(0)+f(1)+f(2)+f(3)+f(4))*1 = 2+6+10+38+114+262 = 432

For 12 rectangles and left endpoints x₁=-1, Δx=0.5

Area of 12 rectangles = (f(-1)+f(-0.5)+f(0)+f(0.5)+f(1)+f(1.5)+f(2)+f(2.5)+f(3)+f(3.5)+f(4)+f(4.5))*0.5

=(2+5.5+6+6.5+10+19.5+38+68.5+114+177.5+262+370.5)*0.5 = 540

C) For 6 rectangles and midpoints x₁=-0.5, Δx=1

Area of 6 rectangles = (f(-0.5)+f(0.5)+f(1.5)+f(2.5)+f(3.5)+f(4.5))*1 = 5.5+6.5+19.5+68.5+177.5+370.5 = 648

For 12 rectangles and midpoints x₁=-0.75, Δx=0.5

Area of 12 rectangles = (f(-0.75)+f(-0.25)+f(0.25)+f(0.75)+f(1.25)+f(1.75)+f(2.25)+f(2.75)+f(3.25)+f(3.75)+f(4.25)+f(4.75))*0.5

=(4.3125+5.9375+6.0625+7.6875+13.8125+27.4375+51.5625+...

+89.1875+143.3125+216.9375+313.0625+434.6875)*0.5 = 657