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Area under Integral.

  • Thread starter azatkgz
  • Start date
  • #1
190
0
This question is quite hard for me.

Homework Statement


Find the area between the curve [tex]y=e^{-x}\left|sinx\right|[/tex] and the straight line y=0 for [tex]x\geq 0[/tex]


The Attempt at a Solution



[tex]\int_{0}^{\infty}e^{-x}\left|sinx\right|dx=-e^{-x}\left|sinx\right|+\int_{0}^{\infty}e^{-x}\left|cosx\right|dx[/tex]
[tex]=-e^{-x}\left|sinx\right|-e^{-x}\left|cosx\right|-\int_{0}^{\infty}e^{-x}\left|sinx\right|dx[/tex]
[tex]\int_{0}^{\infty}e^{-x}\left|sinx\right|dx=\frac{-e^{-x}\left|sinx\right|-e^{-x}\left|cosx\right|}{2}[/tex]
 

Answers and Replies

  • #2
NateTG
Science Advisor
Homework Helper
2,450
5
Because sine and cosine cross the x-axis, you can't just take the absolute value thorough the integral like that.
 
  • #3
190
0
Than ,actually, i should divide it to pieces.
[tex]\int_{0}^{\infty}e^{-x}\left|sinx\right|dx=\int_{0}^{\pi}e^{-x}sinxdx+\int_{\pi}^{2\pi}e^{-x}(-sinx)dx+\cdots[/tex]
[tex]=\sum_{k=0}^{\infty}(-1)^k\int_{k\pi}^{(k+1)\pi}e^{-x}sinxdx[/tex]
 
Last edited:
  • #4
190
0
Is it equals to
[tex](-1)^k\sum_{k=1}^{\infty}\frac{e^{-k\pi}}{2}[/tex] ?
 
  • #5
190
0
Oh,sorry
[tex]\frac{1}{2}+\sum_{k=1}^{\infty}(-1)^ke^{-k\pi}[/tex]
 
  • #6
190
0
from this one I've found [tex]\frac{(1-e^{-\pi})}{2(1+e^{-\pi})}[/tex]



Is this answer true?Please,check.
 
  • #7
Gib Z
Homework Helper
3,346
5
Posts 3 and 5 are definitely correct, though either you or I have made a sign error on the sum of the geometric series, check that again just to be safe.
 
  • #8
190
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I did it in this way
[tex]\frac{1}{2}-\frac{e^{-\pi}}{1-e^{-2\pi}}+\frac{e^{-2\pi}}{1-e^{-2\pi}}[/tex]
 
  • #9
Gib Z
Homework Helper
3,346
5
Your last post makes no sense to me :( Maybe I'm doing the mistake, so heres how I'm getting it

[tex]\frac{1}{2}+\sum_{k=1}^{\infty}(-1)^ke^{-k\pi} = \frac{1}{2} + \frac{-e^{-\pi}}{1+e^{-\pi}} = \frac{1-3e^{-\pi}}{2(1-e^{-\pi})}[/tex]
 
  • #10
190
0
Hey,Gib Z
Should't be your answer be same as mine.
 
  • #11
Gib Z
Homework Helper
3,346
5
lol well i dont know if your making the mistake or me, but i cant see mine, so unless u do, check ur answer again
 

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