Summation of rectangular areas (calculus) problem.

In summary, the problem at hand involves finding the area enclosed between the curve y = x^2 + 2x and the x-axis from x=0 to x=3 using summation of rectangles. To do this, the interval is divided into n equal parts and sigma notation is used to represent the sum of rectangular areas. The main difficulty is isolating the k^2 term in order to write the summation formula and evaluate the limits. However, the problem can be rewritten and simplified in terms of constants and evaluated as n approaches infinity.
  • #1
singleton
121
0
Good evening. I'm having a little difficulty with the summation of rectangular areas when finding the area under a curve.

Question:
Using summation of rectangles, find the area enclosed between the curve y = x^2 + 2x and the x-axis from x=0 to x=3.

Well, I start by dividing the interval (from x=0 to x=3) by n equal parts to find the width of each rectangular area.
=3/n

Then I begin using sigma notation (I'm new at this)
Sum of rectangular areas
[tex]= \sum_{k=1}^n\ f(x) * (3/n)[/tex]
[tex]= \sum_{k=1}^n\ [(k * 3/n)^2 + 2(k * 3/n)] * (3/n)[/tex]
[tex]= \sum_{k=1}^n\ [k^2 * (3/n)^2 + 2 * k * (3/n)] * (3/n)[/tex]
[tex]= \sum_{k=1}^n\ [k^2 * (3/n) + 2k] * (3/n)^2[/tex]
[tex]= 9/n^2\sum_{k=1}^n\ [k^2 * (3/n) + 2k)][/tex]

Now my main problem is that I'm trying to isolate the k^2 so that I can write out the summation formula for it and then go to limits and discover the area under the curve.

eg
[tex]\sum_{k=1}^n\ k^2[/tex]
would become
n(n+1)(2n + 1) / 6
and I could go straight to the limits
 
Last edited:
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  • #2
I'm not sure what your difficulty is - it appears you're right on the mark!
 
  • #3
The difficulty is that I don't know -how- to isolate the k^2 :(

I know very little about using this sigma notation, the only rule I know is that when you have sigma(ak + b) you can rewrite as:
sigma(ak) + sigma(b)
=(a)sigma(k) + (b)sigma(1)

where a and b are constants.

for me, the problem is that there is a variable on the other side of the addition--2k that is...

Again, I know very little of the rules of sigma notation. Perhaps there is a way to rewrite this? I'm not wanting the answer for the area--rather just a more manageable way for me to put it to limits ;)

Or is it already to go? I had just figured I could "do more" to it.

Thanks again!
 
  • #4
Could you experts tell me if what I'm doing below is "legal" in terms of math

[tex]9/n^2\sum_{k=1}^n\ k^2 * (3/n) + 2k[/tex]
becomes
[tex]= 9/n^2[\sum_{k=1}^n\ k^2 * (3/n) + \sum_{k=1}^n\ 2k][/tex]
[tex]= 9/n^2 [(3/n)\sum_{k=1}^n\ k^2 + 2\sum_{k=1}^n\ k][/tex]

And then the first sigma notation would become
n(n + 1)(2n + 1)
--------------
6

and the second sigma notation would become
n(n + 1)
--------
2

and I could then expand the 9/n^2 on it and evaluate the limits? This is probably really stupid question but yesterday was the first day I worked with this notation :D
 
  • #5
Single,

Sorry for the late reply - I lost track of you there!

What you did is fine and you're basically done. Now just ask yourself what the limiting value of your expression is as you let n go to infinity.
 

1. What is the summation of rectangular areas problem in calculus?

The summation of rectangular areas problem in calculus is a mathematical concept where the area under a curve is approximated by dividing it into smaller rectangles and summing their individual areas. This is also known as the Riemann sum, named after the mathematician Bernhard Riemann.

2. How do you find the summation of rectangular areas in calculus?

To find the summation of rectangular areas in calculus, you first need to divide the area under the curve into smaller rectangles. Then, you calculate the area of each rectangle by multiplying its width by its height. Finally, you add up all the individual areas to get the approximate area under the curve.

3. What is the purpose of using the summation of rectangular areas in calculus?

The purpose of using the summation of rectangular areas in calculus is to approximate the area under a curve. This is useful in many real-world applications, such as finding the distance traveled by an object with varying velocity or calculating the total amount of water in a pond with irregular shape.

4. Can the summation of rectangular areas problem be solved exactly?

No, the summation of rectangular areas problem cannot be solved exactly because it is an approximation of the area under a curve. As the number of rectangles used in the calculation increases, the approximation gets closer to the exact value, but it can never be completely accurate.

5. What is the difference between the summation of rectangular areas and the definite integral?

The summation of rectangular areas is an approximation of the area under a curve, while the definite integral is the exact value of the area. The definite integral uses calculus to find the exact area, while the summation of rectangular areas uses simple geometry to approximate the area.

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