How to find the limit process to find the area of the region

Question

How to find the limit process to find the area of the region between f(x)=1/10(x^2+10x) and the x axis on the interval [1,10]

Answer 1

The following graph shows how to divide the target area into 10 rectangles.

 

https://www.desmos.com/calculator/0ayh0k33is

 

If the interval [1,10] is divided into n rectangles all with the same width, then the width of each rectangle will be (10-1)/n or 9/n.

 

Using right hand end-points along the x-axis to get the length of each rectangle results in something like this where x_i is the x-coordinate of that end-point.

x₁= 1 + 9/n   (one width from the starting point)

x₂= 1 + 2(9/n)  (two widths from the starting point)

x_i = 1 + i(9/n) (i widths from the starting point)

 

So a formula for the area of the i^th rectangle is (9/n)[(1/10)(1+9i/n)² + (1+9i/n)] (length times width)

 

So the total area is the ∑ from i = 1 to n of that expression. 

 

That expression can be simplified to a point where you are using formulas like ∑ from i = 1 to n of i = (n)(n+1)/2. There is a similar formula for ∑ i² = [n(n+1)(2n+1)/6]

 

The simplification actually gets really messy. The goal is to get a formula for the total area that involves only n, and then find the limit of that formula as n -> ∞. 

 

You end up with something like this:

 

limit as n-> ∞ [ 729/60 ( 2 + 3/n + 1/n²) + 972/20(1+1/n) + 99/10]

 

Yikes. Picture those fractions with n in the denominator as n gets large getting closer and closer to 0.

 

There is not enough room here to show the work involved to getting to the simplified form.