# 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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### Periodicity of Polar Curves

I am a math educator preparing a unit on the calculus of polar curves. This is my first time teaching this particular unit, so it was also the first time I noticed that the "periods" of ...
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### How to modify off-center circle in polar coordinates so that input angle has a linear relationship with angle on circle?

I have a circle translated horizontally in polar coordinates described by the equation: $$r\left(\theta\right)=d\cos(\theta)+\sqrt{r_{0}^{2}-d^{2}\sin^{2}(\theta)}$$ where $d$ is the horizontal ...
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### Volume of shifted and modified cylinder using polar coordinates

I have a shifted and modified cylinder $x^2+y^2=4x$, bounded below by $z=0$ and above by $z=\sqrt{16-x^2-y^2}$. I want to find its volume. Completing the square and conversion to polar coordinates ...
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### I want to find the volume of a noncircular cylinder that lies inside r=1+$cos(\theta)$ and outside the circle r=1,and top of the cylinder lies on x=z.

First I think as since our $x=z$ then the volume of our right non-circular cylinder's volume is ,where R is the region we integrating and dA is the change in area, $$\int\int_RxdA$$ Then ...
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### Arc length of a polar curve $r = -8\cos(t)$

If I am being asked to find the arc length of the polar curve $r = -8\cos(t)$ when I use the integral formula it gives me $16 \pi$. But since this polar curve represents a circle with radius 4, should ...
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### How to solve this PDE involving polar coordinates?

PDE Equation with Polar Co-ordinates I struggle with using polar coordinates. What method would I use here - is it just the Method of Characteristics? If so, what would be the way to proceed and how ...
1 vote
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### How can i find the average of the product of two functions in polar coordinates? [closed]

I have 2 equations in polar coordinates that are only dependent on $\theta$ and not r: $$f=F(\theta)\\ g=g(\theta)$$ How can I find the product of $<f.g>$. That is the product of the 2 ...
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### What type of spiral is that on the picture ? and what is the formula of such?

I have found some types of spirals, and when I analysed those I have found, they do not met the criteria to shape the draw desired. And a observation point, bacause I think spirograph its a wrong name ...
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### Continuous petal count function

Determine the number of times $r=\sin(nθ),~n$ not necessarily an integer, on a polar coordinate system intersects with a circle centered on the origin with radius 1 for all real numbers. The rose ...
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### Average distance from a point on a circle to the y-axis.

This is a simple question, but I must be making some mistakes as I don't seem to get the answer in the book. Question: Determine the average distance from a point on $x^2+y^2 = 9$ to the $y$-axis. My ...
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### How to find limits of integration when converting to polar coordinates

I'm specifically struggling on finding the integration bounds for $\theta$ as usually the bounds for the radius are clear to me. For example, for the problem $\int\limits_{D} \log(x^2 + y^2) \, dA$ ...
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The question asks $\iint_R (3x+4y^2)\; dA$ where $R$ is the region in the upper half plane bounded by the circles $x^2+y^2=1$ and $x^2+y^2=4$ $$\int_0^\pi \int_1^2 (3r\cos \theta + 3r^2 \sin^2 \theta) ... • 3,185 1 vote 2 answers 54 views ### Intersection of two hyperbolas in polar coordinates. I want to find the intersections of two hyperbolas in polar coordinates. One of their foci coincides, we use this as the pole. (The right focus of the left hyperbola is the same as the left focus of ... 0 votes 1 answer 37 views ### Volume below the cone z=2\sqrt{x^2+y^2} for x^2+y^2\leq4 For x^2+y^2=4,$$z=2\sqrt{4}\Rightarrow z=4$$Since the radius of the basis is 2, then the volume of the cone is$$V=\frac{\pi\cdot2^2\cdot4}{3}\Rightarrow V=\frac{16\pi}{3} However, using ...
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I have to convert the following expression $V(x,y) = x\dfrac{\partial f}{\partial y} - y\dfrac{\partial f}{\partial x}$ to polar coordinates. How do i express the partial derivatives in terms of $r$ ...