Tuesday, 15 November 2016

calculus - Check my proof intinftyinftyintinftyinftysin(x2+y2),dx,dy diverges



I am trying to prove that sin(x2+y2)dxdy diverges and I did it like this:



x=rcosθ, y=rsinθ, θ[0,2π], r[0,], and the jacobian is r:




sin(x2+y2)dxdy=2π00rsin(r2)drdθ=2π0rsin(r2)dr



Use the variable change v=r2, dv=2rdr and so:



2π0rsin(r2)dr=2π0rsin(v)dv2r=π0sin(v)dv



I checked wolfram, and it says that 0sin(x)dx diverges, so I must conclude that the original integral diverges as well.



But how do I actually prove that 0sin(x)dx diverges?


Answer




We can compute the limiting integral by writing 0sin(x)dx=lim We know that the limit \lim_{b\to\infty} \cos(b) diverges. This means the integral diverges as well.


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