Expert: Scotto - 1/8/2010

Question
Estimate the area under the graph
f(x)=4/x from x=1 to x=6 using 5 approximating rectangles and right endpoints. Also it says to repeat this using left endpoints.

My problem is that when I graph f(x)=4/x I don't get set "y" values to find the area of each rectangle, and the area i got of each rectangle going from left to right is for left hand side:
1. 4
2. 2
3. 1.34
4. 1
5. .8

for Right hand side i got going from left to right:
1. 3.98
2. 2.03
3. 1.35
4. .97
5. .8

i got 9.13, which was wrong.

When i added them together i got 9.14 for the left hand side and got it correct, when I drew my rectangles I got sorta the same numbers so i'm not sure what i did wrong or if i drew it wrong.. please help thank you!!  

Answer
Since there are 5 intervals, that means 6 X values,
since there is always one more X values than rectangles.

The left end points would be 1, 2, 3, 4, and 5.
The y values at these points would be 4, 2, 4/3, 1, and 4/5.
When these values are all converted to 15ths, the sum would be 137/15.
Since the interval width for each interval is 1, that is the approximation to the area.
Note that 135/15 = 9, and that is only 2/15 bigger than than 9, and is 9.13, so that's right.

The only error is on the 3, where y should be 1.33 not 1.34, but that's negligable.


The right end points would be 2, 3, 4, 5, and 6.
The y values at these points would be 2, 4/3, 1, 4/5, and 2/3.
Note that the right of ith interval is the same as the left of the (i+1)th interval.
Converting all of these numbers to 15ths and adding them gives 87/15, which is 29/5, or 5.8.
Since the interval width for each interval is 1, that is the approximation to the area.