Need Help please!!

[flajoton]

I have this problem for homework and i do not know how to do it. The problem states:
Find an expression for the area under the graph of "f" as a limit. Do not evaluate the limit.

f(x)= ln(x)/x, 3<x<10

Note: what i do not understand is that I am supposed find this expression by using right end points. (Rn)
Also the less than signs are supposed to be less than or equal to. I just did not know how to do that.
Thank you very much for your time.

 

[galactus]

I assume this is what you are getting at:

Each subinterval will have length \(\displaystyle {\Delta}x=\frac{b-a}{n}=\frac{10-3}{n}=\frac{7}{n}\).

Where n is the number of subintervals we are breaking it up into.

The right endpoint method is \(\displaystyle x_{k}=a+k{\Delta}x=3+\frac{7}{n}\cdot k\)

Thus, each kth rectangle has area \(\displaystyle f(x_{k}){\Delta}x=\left(\frac{ln(3+\frac{7k}{n})}{3+\frac{7k}{n}}\right)\cdot \frac{7}{n}\)

Adding up the areas of all the rectangles, as the number of them heads off toward infinity, gives the area under the curve.

See, as n gets larger and larger, the area of the kth rectangle gets smaller and smaller. Thus, getting closer and closer to the actual area under

the curve.

That is essentially what we are doing when we integrate. Adding up the area of an infinite number of rectangles.

Well, sometimes the area of trapezoids and other times more like parabolas (Simpson's rule).

The area of a rectangle is width times height. In this case, width is \(\displaystyle \Delta x\) and \(\displaystyle f(x)\) is the height.

Thus, the area is \(\displaystyle f(x)\cdot {\Delta}x\)

But, we have to add them all up

\(\displaystyle \sum_{k=1}^{n}f(x_{k}){\Delta}x\)

The sum tells us we are adding up the area of all the rectangles. The limit means we are making the number of them get larger and larger

\(\displaystyle \lim_{n\to {\infty}}\sum_{k=1}^{n}\frac{7ln(\frac{7k+3n}{n})}{7k+3n}\)