Estimate the area under the graph

Question

Suppose you estimate the area under the graph of f(x)=x3 from x=3 to x=23 by adding the areas of rectangles as follows: partition the interval into 20 equal subintervals and use the right endpoint of each interval to determine the height of the rectangle.
What is the area of the 17th rectangle?

Answer 1

You will be dividing the segment along the x-axis from 3 to 23 into 20 equal subintervals, so each subinterval with have a length of (23-3)/20 = 1. 

 

Now the question is what will be the x-coordinate of the right endpoint of the 17th rectangle?

 

Let's look for a pattern.

 

We can find the right endpoint of the first rectangle by starting at 3 and adding 1 width where the width is the length of a subinterval (so 3+1(1) = 4).

 

Similarly to find the right endpoint of the 2nd rectangle start at 3 and add on 2 widths (3+2(1)) = 5. In other words this is an arithmetic sequence. The right endpoint of the 17th rectangle is 3 + 17(1) = 20.

 

f(20) = 20³=8000, which represents the length of the 17th rectangle.

 

The area of the rectangle is lw: 8000(1) = 8000 square units.