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Calculus/area under a curve

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Expert: - 11/16/2011


Question
Scotto!

I have searched the entire internet to answer this simple question: what is the TOTAL area under the curve y= 1/x^2 (combining the areas under the curve from the origin to infinity to the right AND left of the Y-axis)?? Is it 2* pi^2/6? or 4* pi^2/6 + 2? Or something else? I know it is convergent.

It has been 37 years since I took Calculus in college. All of the internet sites want to retrain me, or express only part of the TOTAL AREA or just give hints.

Thanks in advance.

Lorne

Answer
So, 2011 - 37 = 1972, and that is when my oldest brother started college,
but I was coming out of 8th grade.  It seems like that was when Nixon president ...

To find the total area, we need to integrate 1/x² from infinity down to 0.
To integrate x to a power, add 1 to the power and divide by the new power.
Since the power is -2, -2 + 1 = -1, so ∫1/x² dx = -1/x.

At infinity this is known to be 0.
When 0 is put in, however, the result is -1/0, which is infinity, so this is not integrable.

Since the curve is a square, we know that the values on the left side of the axis are the same as the values on the right side of the axis.  Since it is non-intregable on one side, it is non-integrable on the other.

If the integral was from 1 to infinity, the value of it would be -0+1 = 1.
If it was integrated from -infinity to 1, the value of the integral would still be 1.
Between -1 and 1, however, the area is unbounded.

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Scotto

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Any kind of calculus question you want. I also have answered some questions in Physics (mass, momentum, falling bodies), Chemistry (charge, reactions, symbols, molecules), and Biology.

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Experience in the area: I have tutored students in all areas of mathematics for over 25 years. Education/Credentials: BSand MS in Mathematics from Oregon State University, where I completed sophomore course in Physics and Chemistry. I received both degrees with high honors. Awards and Honors: I have passed Actuarial tests 100, 110, and 135.

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