Substitution in integrals

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Generated from the video's visuals and explanation; not verbatim speech.

0:00The screen splits into two views: a standard Cartesian coordinate system ($xOy$) on the left and a parameter plane defined by angle $\theta$ and radius $r$ on the right. An animation draws a closed disk centered at the origin with radius 1 (labeled $r=1$) on the left grid. Simultaneously, a red rectangle appears on the right, spanning horizontally from 0 to 1 and vertically from 0 to $2\pi$. This visualizes how a complex curved boundary transforms into a simple rectangular domain in the new variables.

0:14Above the graphs, the polar coordinate transformation equations appear within braces: $x = r \cos \theta$ and $y = r \sin \theta$. Immediately following this, the Jacobian determinant is displayed: $J = \begin{vmatrix} \cos \theta & -r \sin \theta \\ \sin \theta & r \cos \theta \end{vmatrix}$. The calculation simplifies directly to $= r$, demonstrating the necessary scaling factor required when converting area elements between these systems.

0:20A change of variables updates the integrand, the region and the area factor |J|=r together. The integral remains two-dimensional; the description may simply become easier to use.

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