Unraveling the Mystery of Integration: Area Under the Curve

In summary, the area under the curve is just the result of multiplying the original function by (x/n). This happens to be the area under the curve because it is the only area that meets the requirements of the equation. The area of the rectangle that is created is not the total area under the curve, it is just the area under the curve from 0 to x.
  • #1
BBRadiation
10
0
I am having trouble figuring out how an integration works.

Consider the function, y = x2 . The antiderivative is x3 / 3 or in other words its x2 (x/3). How come multiplying the original function by (x / n) (where n represents the exponent + 1) will lead to the area under the curve.

Put in another way, let's define the above in terms of a graph. Consider the function y = x2 and you want to find the area from 0,1.

All you are doing to find the area is x2 (x/3) . You are creating a rectangle in which the dimensions are x2 by (x/3). If the function we were dealing with were x3, the rectangle that would yield the area under the curve would be x3 by ( x / 4)

http://img237.imageshack.us/img237/6906/image1sj.png [Broken]

How come the above rectangle just happened to be the total area under the curve? How did this work out and why does it work out?

In order to try and figure this out, I searched the history of calculus and found this from wiki, but I didn't really understand it:
In this paper, Newton determined the area under a curve by first calculating a momentary rate of change and then extrapolating the total area. He began by reasoning about an indefinitely small triangle whose area is a function of x and y. He then reasoned that the infinitesimal increase in the abscissa will create a new formula where x = x + o (importantly, o is the letter, not the digit 0). He then recalculated the area with the aid of the binomial theorem, removed all quantities containing the letter o and re-formed an algebraic expression for the area. Significantly, Newton would then “blot out” the quantities containing o because terms “multiplied by it will be nothing in respect to the rest”.
 
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  • #2
If you read any standard calculus book, you'll see how formulas for areas under such curves are derived, and why they are true.

The equality of the areas of the region you have in mind and the area of that rectangle happen to be a "fun fact", I don't think there's a way to show this geometrically, it's just something that we can observe from the formula we get for the area under the curve x^n.

Also, the area of the rectangle is not the total area under the curve, it's area under the from 0 to x.
 
  • #3
If you read any standard calculus book, you'll see how formulas for areas under such curves are derived, and why they are true.

Do you have any specific books in mind?

Also, the area of the rectangle is not the total area under the curve, it's area under the from 0 to x.

Sorry, that's what I meant.

I am just having trouble understanding why multiplying the original function by (x/n) can lead to the area under the curve.
 
  • #4
BBRadiation said:
I am just having trouble understanding why multiplying the original function by (x/n) can lead to the area under the curve.

It's a coincidence. Kinda like how sum of the first n odd numbers is n^2 or how
[tex]
1^3 + 2^3 + 3^3 + ... = (1 + 2 + 3 + ...)^2
[/tex]

We can prove these formulas are true, but the fact that they are true are very neat coincidences.
 
  • #5
http://www.sci.uidaho.edu/polya/math170/modules/?mod=14&sec=1&sub=0 [Broken]

Watch the Reimann sums video @ this link. It's short and will convey to you the general idea of what is going on.

This video is then extremely worth watching afterwards, (actually any videos by this guy is worth watching!),
http://www.5min.com/Video/Areas-Riemann-Sums-and-Definite-Integrals-169053988

Then, if you'd then like to know more, keep watching the videos and there's also some great videos @ www.khanacademy.org to go a little deeper and get a good textbook on the subject. Btw, don't bother with wikipedia it's terrible for beginning to understand a lot (but not all) of math stuff and will scare you away because of the way it jumps from simple to mind bogglingly difficult to simple on about every math topic...
 
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What is the concept of integration?

The concept of integration, also known as finding the area under the curve, is a way to calculate the total value of a function over a specific interval. It involves breaking down a curved or irregular shape into smaller, simpler shapes, calculating the area of each shape, and then adding them together to get the total area.

What is the purpose of integration?

The purpose of integration is to find the total value of a function over a specific interval. This can help in various fields such as physics, engineering, and economics where calculating the total value of a changing quantity is necessary. Integration also allows us to find important information about a function, such as its average value and maximum and minimum values.

What are the different methods of integration?

There are several methods of integration, such as the Riemann Sum method, the Trapezoidal Rule, and Simpson's Rule. These methods involve dividing the interval into smaller subintervals and using different mathematical formulas to approximate the area under the curve. Additionally, integration can also be done using anti-derivatives, which involve finding the inverse function of the original function.

What are the applications of integration in real life?

Integration has many real-life applications, such as calculating the distance traveled by a moving object, finding the volume of irregularly shaped objects, and determining the total revenue or profit of a business. It is also used in fields such as economics, biology, and statistics to analyze data and make predictions.

What are some common challenges faced while solving integration problems?

One common challenge in solving integration problems is determining the limits of integration, which define the interval over which the function is being integrated. Additionally, some functions may be difficult to integrate using traditional methods and may require advanced techniques such as substitution or integration by parts. It is also important to pay attention to the order of operations and use the correct formulas for each type of integration problem.

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