Estimate 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

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

A). Using Right Endpoints

With 6 rectangles the width of each rectangle will be: (5 - -1)/6 = 1

The heights will be dependent on the input on the right side of each rectangle, so you will need to evaluate the function at x-values of 0, 1, 2, 3, 4, 5

f(0) = 6

f(1) = 10

f(2) = 38

f(3) = 114

f(4) = 262

f(5) = 506

Then sum the area of each rectangle. The Area of a given rectangle can be found by width * height

1(6) + 1(10) + 1(38) + 1(114) + 1(262) + 1(506) = 936

------------------------------------------------

With 12 rectangles the width of each rectangle will be: (5 - -1)/12 = 0.5

The heights will be dependent on the input on the right side of each rectangle, so you will need to evaluate the function at x-values of -0.5, 0, 0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, 5

f(-0.5) = 5.5

f(0) = 6

f(0.5) = 6.5

f(1) = 10

f(1.5) = 19.5

f(2) = 38

f(2.5) = 68.5

f(3) = 114

f(3.5) = 177.5

f(4) = 262

f(4.5) = 370.5

f(5) = 506

Then sum the area of each rectangle. The Area of a given rectangle can be found by width * height

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

If you want more help with Parts B and C, please contact me for some one-on-one tutoring.