# How do I estimate the area under the graph of #f(x) = sqrt x# from x=0 to x=4 using four approximating rectangles and right endpoints?

##### 1 Answer

graph{sqrt(x) [-5, 5, -2.5, 2.5]}

Look at the graph above.

Divide the interval into the correct number of equal pieces. In this case, the interval is [0,4], and their are 4 pieces. so:

That is, each interval should be 1 unit wide.

Another way of saying this would be that if I break the x-axis between 0 and 4 into 4 equal segments, each segment would be 1 unit long. (see the image below:

Now, the right endpoints are x=1, 2, 3, and 4.

(the left endpoints are x=0, 1, 2, and 3)

Making rectangles from those endpoints looks like this:

Now to find the area under the curve, using the rectangles is simply Area = Base * Height. In this case, the base of each rectangle is 1, and the height is

I'll let you do the math.

I will let you know these things, though (a quick look ahead):

1) Using the right side overestimates the area

2) Using the left side underestimates the area

3) using the *middle* is much closer, but is not exact, either

4) If you increase the number of rectangle while holding the interval of interest the same (say 1,000 rectangles from 0 to 4), it becomes more accurate.

5) the maximum number of rectangles is the *limit* of the number of rectangles

6) as the limit approaches infinity, the answer approaches *exact*, and is called an integral.