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Integration of e

  • Thread starter Jovy
  • Start date
  • #1
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2

Homework Statement


[/B]

$$y= \frac 1 {\sqrt{2\pi}}e^{ \frac{- x^2} 2},~y=0,~x=0,~x=1$$

Homework Equations



$$Volume=2\pi\int_a^b p(x)h(x)dx$$

The Attempt at a Solution



I understand how to do the problem, I'm just having trouble integrating.
##h(x)=\frac 1 {\sqrt{2\pi}}e^{ \frac{- x^2} 2},~ p(x)=x##
And you just plug that into the equation.
When you integrate ##\frac 1 {\sqrt{2\pi}}e^{ \frac{- x^2} 2}x## , you use the chain rule right? But first you move the constant ##\frac 1 {\sqrt{2\pi}}## out of the integral next to the ##2\pi##
So it would look like this, ##Volume=2\pi~\frac 1 {\sqrt{2\pi}}~\int_0^1~ xe^{ \frac{- x^2} 2}dx##

I know the anti derivative of ##e^{ \frac{- x^2} 2} is, e^{ \frac{- x^2} 2}## and then you have to do the anti derivative of ## { \frac{- x^2} 2}## as part of the chain rule right? Well, that is where I get stuck.
 

Answers and Replies

  • #2
Buzz Bloom
Gold Member
2,211
365
Hi Jovy:

I suggest you investigate the substitution u = (x^2)/2, du = x dx

Hope this helps.

Regards,
Buzz
 
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  • #3
Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
10,706
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Homework Statement


[/B]

$$y= \frac 1 {\sqrt{2\pi}}e^{ \frac{- x^2} 2},~y=0,~x=0,~x=1$$

Homework Equations



$$Volume=2\pi\int_a^b p(x)h(x)dx$$

The Attempt at a Solution



I understand how to do the problem, I'm just having trouble integrating.
##h(x)=\frac 1 {\sqrt{2\pi}}e^{ \frac{- x^2} 2},~ p(x)=x##
And you just plug that into the equation.
When you integrate ##\frac 1 {\sqrt{2\pi}}e^{ \frac{- x^2} 2}x## , you use the chain rule right? But first you move the constant ##\frac 1 {\sqrt{2\pi}}## out of the integral next to the ##2\pi##
So it would look like this, ##Volume=2\pi~\frac 1 {\sqrt{2\pi}}~\int_0^1~ xe^{ \frac{- x^2} 2}dx##

I know the anti derivative of ##e^{ \frac{- x^2} 2} is, e^{ \frac{- x^2} 2}## and then you have to do the anti derivative of ## { \frac{- x^2} 2}## as part of the chain rule right? Well, that is where I get stuck.
No, the antiderivative of ##e^{-x^2/2}## is a non-elementary function. In other words, it is not possible to write down the antiderivative in a finite number of terms involving the standard functions. Even if you try to write it down in 1 billion pages of horrible algebra, you cannot do it.

Fortunately, you don't need the antiderivative of ##e^{-x^2/2}##, but of ##x e^{-x^2/2}## instead, and that is a much different story.
 
  • #4
33,632
5,287

Homework Statement


[/B]

$$y= \frac 1 {\sqrt{2\pi}}e^{ \frac{- x^2} 2},~y=0,~x=0,~x=1$$
What exactly is the problem? All you have given us is an equation in x and y, but you haven't said what it is you're supposed to do?

From your attempt below, it would seem that you're trying to find the volume of a surface of revolution, but there is no clue above that that's what you need to do.
Jovy said:

Homework Equations



$$Volume=2\pi\int_a^b p(x)h(x)dx$$

The Attempt at a Solution



I understand how to do the problem, I'm just having trouble integrating.
##h(x)=\frac 1 {\sqrt{2\pi}}e^{ \frac{- x^2} 2},~ p(x)=x##
And you just plug that into the equation.
When you integrate ##\frac 1 {\sqrt{2\pi}}e^{ \frac{- x^2} 2}x## , you use the chain rule right? But first you move the constant ##\frac 1 {\sqrt{2\pi}}## out of the integral next to the ##2\pi##
So it would look like this, ##Volume=2\pi~\frac 1 {\sqrt{2\pi}}~\int_0^1~ xe^{ \frac{- x^2} 2}dx##

I know the anti derivative of ##e^{ \frac{- x^2} 2} is, e^{ \frac{- x^2} 2}## and then you have to do the anti derivative of ## { \frac{- x^2} 2}## as part of the chain rule right? Well, that is where I get stuck.
 

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