Are all the regions intermediary between the initial one and the final one also of equal area ? Or just the initial and final one ?
Every time you use u-sub, you're actually transforming your region into a different one with equal areapic.twitter.com/hFyRPqd9wb
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great question. I don't believe so. The way i wrote the code, i just came up with a sequence of functions that smoothly transitioned to cos(x).. so no unless i got lucky
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That's the point of substitution, no? That we can solve a simpler integral of the provably same value instead?
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indeed! but many people don't usually get taught that you're actually transforming a region. As you can see in the animation, the fact that the areas are equal isn't obvious geometrically
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Just substitute u=x^2, in fact all possible subtitutions like u=e^x etc. transform an integral in such a way that the area is preserved i.e. an invariant. This can also be shown for multi dimensional/multiple integrals wherea Jacobian /Jacobi ("J") determinant is used. -
this is precisely what it did.. the purpose of this video is giving an intuition for what that substitution is actually doing..
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Except when your u-sub has a discontinuity on the interval.
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No it is valid again.
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I actually didn’t realize this. Very cool!
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It’s a lot more obvious in multiple dimensions, I think—I similarly hadn’t made the mental connection to 1d
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