Estimate Area

Question

Estimate the area under the graph of f(x) = 1/(x+4) over the interval over the interval [0,5] using eight approximating rectangles and right endpoints.

 

R_n= ?

 

Repeat the approximation using left endpoints.

 

L_n= ?

Answer 1

The width of each subinterval/rectangle is (5-0)/8 = 5/8.

To find x₁,x₂,...x₈ using righthand endpoints, start at 0 and add on 1 width to get to x₁. Then keep adding on widths until you get to x_8.

^x₁ = 0 + 1(5/8) = 5/8

x₂ = 5/8 + 5/8 = 10/8

x₃ = 10/8 + 5/8 = 15/8

and so on... 20/8, 25/8, 30/8, 35/8, 40/8.

Those values can be reduced, but for now let all the denominators remain 8.

The area of the 1st rectangle is length times width.

A₁ = (5/8)f(5/8) = (5/8) (1/ [(5/8) + 4]) =(5/8)(8/37)

A₂ = (5/8)f(10/8) = (5/8) (1/[(10/8) + 4]) = (5/8)(8/42)

A₃= (5/8)f(15/8) = (5/8)(1/[(15/8) + 4]) = (5/8)(8/47)

...

R₈ = (5/8)[8/37 + 8/42 + 8/47 + 8/52 + 8/57 + 8/62 + 8/67 + 8/72] ≈ 0.76916

For lefthand x_i:

x₁ = 0

x₂ = 5/8

x₃ = 10/8

...

A₁ = (5/8)f(0) = (5/8)(1/4)

A₂ = (5/8)f(5/8) = (5/8)(8/37)

A₃ = (5/8)f(10/8) = (5/8)(8/42)

...

L₈ = (5/8) [1/4 + 8/37 + 8/42 + 8/47 + 8/52 + 8/57 + 8/62 + 8/67] ≈ 0.85596

This Desmos graph does the calculations:

desmos.com/calculator/ix6ktk2ooe

Use the slider on "c" (choice) to choose left, right , or midpoint.