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   Compare an Integral and a Number		 « Previous topic | Next topic »
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   Author  Topic: Compare an Integral and a Number  (Read 464 times)
Barukh
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Compare an Integral and a Number  
« on: Aug 11th, 2004, 10:12am »		 Quote Quote Modify Modify
Which number is bigger?
 
1. [int]0[pi] e^(sin2x) dx
 
2. 3[pi]/2
 
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Aryabhatta
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Re: Compare an Integral and a Number  
« Reply #1 on: Aug 11th, 2004, 11:34am »		 Quote Quote Modify Modify
let f(x) = e^sin[sup2]x
Now f(x) [ge] 1 + sin[sup2]x + (sin4x)/2
 
So  
[int]0[pi]f(x)dx > [int]0[pi](1 + sin[sup2]x)dx = 3[pi]/2
 
as [int]0[pi](sin4x)/2 > 0 and
 
[int]0[pi]sin[sup2]xdx = 2[int]0[pi]/2sin[sup2]xdx = 2[int]0[pi]/2cos[sup2]xdx = [int]0[pi]/2(sin[sup2]x + cos[sup2]x)dx = [pi]/2
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