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I don't know where is a contradiction to my understanding with the following argument. Can anyone help me to point out my mistake.

We know that the co-area states as follows (everything in $\mathbb{R}^{2}$):

$$ \intop_{\left\{ x^{2}+y^{2}<1\right\} }f\left(x,y\right)dxdy=\intop_{0}^{1}dt\intop_{\sqrt{x^{2}+y^{2}}=t}f\left(x,y\right)dS\left(x,y\right). $$ Now look at the right-hand side, we can write $$ \intop_{0}^{1}dt\intop_{\sqrt{x^{2}+y^{2}}=t}f\left(x,y\right)dS\left(x,y\right)=\intop_{0}^{1}dt\intop_{0}^{2\pi}f\left(t\cos\theta,t\sin\theta\right)d\theta. $$ However, using polar coordinate formula, the left-hand side is $$ \intop_{\left\{ x^{2}+y^{2}<1\right\} }f\left(x,y\right)dxdy=\intop_{0}^{1}\intop_{0}^{2\pi}f\left(t\cos\theta,t\sin\theta\right)\times\underbrace{t}_{Jacobian}\times d\theta dt. $$

The contradiction follows by the fact that the last two terms are not equal. Where is my mistake?

Thanks.

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The arclength element $dS$ (in your notation) is $t\cdot d\theta$, not simply $d\theta$.

(Remember that the length of an arc subtended by angle $\theta$ in a circle with radius $r$ is $r\theta$.)

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