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how to write this region $D$ in relation to $r,\theta$ in this $\iint_Df(x,y)dxdy$ where $D=\{x^2+y^2 \le1,x+y\le 1\}$ and $D=\{x^2+y^2\le1,x+y\ge1\}$

In the case $D=\{x^2+y^2 \le 1,\, x+y\ge 1\}$, you have $0\le r\le1$ and $r(\cos \theta+\sin \theta)\ge 1$. The angle $\theta$ varies between $0$ and $\pi/2$, so the integral becomes: $$\iint_D f(x,y)...
Sine of the Time's user avatar
1 vote
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Is there a way to derive the Polar Curve Area Formula using Parametrics?

The mapping of the Cartesian to Polar coordinates is $x=r\cos\theta$, $y=r\sin\theta$. The transformation of areas is $$ \int\int dx dy = \int\int ||\begin{array}{cc} \partial x/ \partial r & \...
R. J. Mathar's user avatar
  • 2,855
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Area of a cardioid and a circle

It's not the first half of the circle but the full circle. The origin lies on the circumference, not at the centre. All of the circle lies between the polar arguments $0$ and $\pi.$
Lieven's user avatar
  • 1,918

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