Wednesday, May 4, 2016

integration - What does intsin(dx) mean?

This is only about, how I came up with this weird idea.




I was considering about relationship between radius(r),arc of the circle (s) and radian (θ) such that;



enter image description here



r/s=θ




And to convince myself completely, I wanted to try following method.
When s,ϕ are very small like δs and δϕ, we can assume that the triangle that rsr with angle δϕ that is;
enter image description here



So from the triangle, we can conclude that sin(δϕ2)=δs2r and while δs,δϕ go to the 0 we can take integral at both side.




Conclusion:




What does following mean?



I=sin(dx)



1.



I considered that what if we take the integral to inside of sin(x)?



so I=sin(x+C)




2.



I've tried definition of riemann integral.



baf(x)dx=limnnk=0f(a+kban)(ban)



But what is the function? f(dx) doesn't look like just f(x) or I can try following but it doesn't make any sense to me, as well.



f(dx)=f(dx)dxdx so U(x)=f(dx)dx, but I couldn't finish.

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