Another problem

[Lizzie]

The problem:
If f(x) = 3x - 7 , 0 is less than or equal to x which is less than or equal to 3 , evaluate the Riemann sum with n = 5 , taking the sample points to be right endpoints. What does the Riemann sum represent? Illustrate with a diagram.

Where I am at . . . :
I plugged the equation into my calculator and used the following window : xmin=0, xmax=3, xscl=1, ymin=-10, ymax=5, yscl=1, xres=1. I see that until x reaches 2 and 1/3, the graoh is below the x axis, then it reaches above. By checking by calc book, I found that I am supposed to find the approximate area of the above part and subtract the approximate area of the below part. Ok, so then what do I do with the n = 5 thing? Once again, I may have missed something. Any help is greatly appreciated, as it always is!

 

[galactus]

This is essentially the same as the other problem you posted.

Find the change in x, \(\displaystyle \frac{3-0}{5}=\frac{3}{5}\)

What you have to do is find the height of the right most part of the rectangle.

You get that by plugging in the appropriate values.

This particular problem has 5 partitions, so the rightmost part of the first rectangle

will be at x=3/5, plug that into y=3x-7 to get the height. \(\displaystyle 3(\frac{3}{5})-

7=\frac{-26}{5}\). Height times width equals

area. See.? Now you do it again. Go to the second rectangle. \(\displaystyle (\frac{3}{5})+

(\frac{3}{5})=\frac{6}{5}\)

Now, plug 6/5 into your equation, \(\displaystyle 3(\frac{6}{5})-7=\frac{-17}{5}\)

Keep adding 3/5 until you get to the end of your interval.

Here are the first two. Can you find the other three and the total area?.

(3(3/5)-7)+(3(6/5)-7)+?+?+?=?

These are all the heighths or y values. Then, multiply the total by the subinterval,

3/5, or the x.

That gives area. See?. It's actually a wonderful concept which shows how calculus

works. You're adding up the area of all the rectangles. Of course, the more

rectangles the closer the area. Hence, integral calculus.

Here's another graph to show you. I want you to be able to see what is going on

here.

 

[Lizzie]

First, thank you so much for your wonderful explanation and graph. I really respond well to visual aids.
Next,
I'm guessing that if the first two are (3(3/5)-7)+(3(6/5)-7), then it continues like this: +(3(9/5)-7)+(3(12/5)-7)+(3(3)-7), which would equal -8. Then do I just multiply it by 3/5?

And just so I understand what you are saying, if I have this info: r is less than or equal to x which is less than or equal to s and n = t, then I can translate that to (s-r)/n as the change in x?

 

[stapel]

I was replying earlier, but dinner interrupted before I could click "Submit". Your question has been answered (and very nicely, galactus!), so all that's left for me to post is what had been a post-script:

What calculator are you using? I've written a Riemann-sums program for my TI-84 that does left endpoints, right endpoints, and midpoints, and the program should work for TI-83's, too. Send me a private-message if you're interested.

Eliz.

 

[galactus]

First, thank you so much for your wonderful explanation and graph. I really respond well to visual aids.
Next,
I'm guessing that if the first two are (3(3/5)-7)+(3(6/5)-7), then it continues like this: +(3(9/5)-7)+(3(12/5)-7)+(3(3)-7), which would equal -8. Then do I just multiply it by 3/5?

And just so I understand what you are saying, if I have this info: r is less than or equal to x which is less than or equal to s and n = t, then I can translate that to (s-r)/n as the change in x?

By Jove, I think she's got it!.....Now, pretend your Audrey Hepburn and sing a score from "My Fair Lady". :lol: I'm glad I could help. I'd take Stapel up on the offer.

 

[Lizzie]

I sure did take her up on the offer! lol, I'm no dummy! *thinks of a tune to sing*

 

[Lizzie]

OK, so my final answer should look something like this?

A=-4.3 units²
The Riemann sum represents an equation that, between x=0 and x=3 is under the x axis -4.3 units² more than it was above the x axis. *is this correct?*
And then I'd draw the diagram of the Riemann sum?

 

[galactus]

Almost. \(\displaystyle \frac{3}{5}(-8)=\frac{-24}{5}=-4.8\)

 

[Lizzie]

lol, I meant to write -4.8, what a dummy! Glad you caught that, btw.