Find the area under the curve using right endpoints and left endpoints

Question

Estimate the area under the graph of ƒ(x)=1/(x+1) over the interval (0,3) using four approximating rectangles and right endpoints.

Find R_n

Repeat the approximation using left endpoints.

Find L_n

Thank you!

Answer 1

For the right endpoints the picture looks like this:

The rectangles divide the length "3" into 4 equal pieces so each rectangle is 3/4 units wide (the x values at each rectangle break occur at x = 0, x = 0.75, x = 1.5, x = 2.25, and x = 3)

The first rectangle has a height of f(0.75) or 1/(0.75 + 1) = 1/1.75 = 4/7

The second rectangle has a height of f(1.75) or 1/(1.5 + 1) = 1/2.5 = 2/5

The third rectangle has a height of f(2.25) or 1/(2.25 + 1) = 1/3.25 = 4/13

And the fourth rectangle has a height of f(3) or 1/(3 + 1) = 1/4

So the area R_n = 0.75(4/7 + 2/5 + 4/13 + 1/4) ≈ 1.14684

Notice that this is an underestimate since we are missing area that the rectangles are not covering.

The same holds true for L_n expect now the first rectangle has a height of f(0), the second has a height of f(0.75) etc. The result is: 0.75(1 + 4/7 + 2/5 + 4/13) ≈ 1,70934

This would be an overestimate since the rectangles will be above the function graph line. The true are is somewhere in the middle.