# Questions tagged [polar-coordinates]

Questions on polar coordinates, a coordinate system where points are represented by their distance from the origin ($r$) and the angle the line joining the point and the origin makes with the positive horizontal axis ($\theta$).

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### Bound for polar equations.

When I wish to find the area of a polar equation such as $r=2-2\cos\theta$, I need to compute $$\frac12\int_\alpha^{\beta}r^2.$$ However, I am confused as to how to determine $\alpha$ and $\beta$. I ...
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### Evaluating $\int_{0}^1\int_{0}^{\sqrt{4y-y^2}}x^2dxdy$ using polar coordinates

Evaluate $$\int_{0}^1\int_{0}^{\sqrt{4y-y^2}}x^2dxdy$$ using polar coordinates. My try: The region is as shown below: As i can notice $\theta$ is going from $0$ to $\frac{\pi}{2}$. But i am confused ...
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### The points A $(0,0)$, B$(\cos(\alpha),\sin(\alpha))$ and C $(\cos(\beta),\sin(\beta))$ are the vertices of a right angled triangle.

$(0,0)$, B$(\cos(\alpha),\sin(\alpha))$ and C $(\cos(\beta),\sin(\beta))$ are the vertices of a right angled triangle. Derive a relation between $\alpha$ and $\beta$." /> I tried using the slope ...
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### Converting simplified Gielis equation (polar) to parametric equation

I found the simplified Gielis equation, that discribes the shape of a leaf. The equation states: $r = \frac{l} {(|cos \frac{θ}{4}| + |sin\frac{θ}{4}|)^\frac{1}{n}}$ To get to the Cartesian ...
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### Spiral enlargements on desmos

If you toggle the slider for n the point moves in a spiral. Is there a way to plot the curve for all values of n? https://www.desmos.com/calculator/kmdmfazsyy
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### Rotating polar velocity vector fields

There is a great way to rotate a Cartesian vector field about the origin described in Rotating vector functions. Instead, let us suppose that we have a velocity vector field in polar coordinates i.e., ...
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### How does the ellipsis $x^2+2y^2=2$ gets represented to $x=\sqrt{2}\cos\theta; y= \sin(\theta)$ in polar coordinates?

Just like the title says, How does the ellipsis with equation $$x^2+2y^2=2$$ becomes represented as $$x=\sqrt{2}\cos\theta; y= \sin(\theta)$$ in polar coordinates? can someone help me to understand ...
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### Express Laplacian in polar coordinates

Part of this problem is asking to express $u$ in polar coordinates and express the domain and BCs to those coordinates. The PDE is the Laplacian on disc with BC $u=0$: $\Delta u+\lambda u=0, \quad$ ...
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### Length of a line to the surface of a sphere when line does not originate from center

Finding Length of a line to the surface of a sphere when line does not originate from center. Here is what I know: I have a 3D sphere of radius $7$ I have a line (...
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### Double and triple integral in spherical polar coordinates, $\iint\sqrt{4-x^2-y^2}\mathrm{d}A$
Consider the double integral $$I=\iint\limits_{\mathcal D} \sqrt{4-x^2-y^2}\mathrm{d}A$$ where $\mathcal D=\{(x,Y): x^2 + y^2 \leq 4\}$ is the disc on the $xy$ plane (source) A.) Use polar ...