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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Polar Coordinates

I dont really know how to go about this question. I know that the area is $$\int_\alpha^\beta \frac12 r^2\, d\theta $$ The question is to find the area of the shaded region.
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5answers
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Points on $(x^2 + y^2)^2 = 2x^2 - 2y^2$ with slope of $1$

Let the curve in the plane defined by the equation: $(x^2 + y^2)^2 = 2x^2 - 2y^2$ How can i graph the curve in the plane and determine the points of the curve where $\frac{dy}{dx} = 1$. My work: ...
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3answers
85 views

Area and Polar Coordinates

Would anyone be able to help me with this problem? I think I know the area formula in polar coordinates that should be used: the antiderivative of ((1/2)r^2 dtheta) from alpha to beta but I'm not ...
5
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1answer
47 views

Polar Coordinates — Equation of a line

Hey, does anyone know how to tackle this question? I've tried using the formula r=d*sec(theta-alpha)but I'm not sure what each of the variables are equal to. If anyone can offer any help, it would ...
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1answer
55 views

Polar to cartesian form of $ r = \sin(2\theta)$

As title describes, I was wondering how I would put this into cartesian form, from polar. All I have is $ r = \sin(2\theta)$. I'm not really sure what to do, i've been trying to find similar ...
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1answer
26 views

Polar equation for a Heptagon

What is polar equation for a Heptagon ? I need to move some Android views in the form of a heptagon, for i need to have polar equations for Heptagon like for x= rSine(Theta) and y=rcos(theta). is it ...
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1answer
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I need help converting $x^2 + y^2 = -4y$ into a polar equation

I know the whole $r^2 = x^2 + y^2$ and $x = r \cos \theta$ and $y = r \sin \theta$, but I just can't seem to apply those rules to the equation $x^2 + y^2 = -4y$ to make it a polar one.
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1answer
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Relationship between two perpendicular lines expressed in polar coordinate?

I've been studying Hough transform. Basically, let's say we have a line $$y = mx+b$$ We can change our view to a parametric view (e.g. parameter space of $m,b$ while $(x,y)$ is constant). This would ...
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1answer
37 views

Double Integral Help $(x^2+y^2+a^2)^{-2} dx \, dy$

Hi I'm currently revising for a maths module that I am taking as part of my physics degree. All was going well until I hit a dead-end with this integral, any ideas how to evaluate it? $$ ...
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1answer
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trying to figure out how to set up polar area given two equations

R=3 and R=4-2sin(Theta) Find the area I got the second part of the equation right (1/2 the integration of pi/6 to 5pi/6 of (4-2sinx)^2) but he first part confuses me. The answer key stated that ...
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1answer
175 views

Find the area of the region that lies inside both curves $r = 5 \sin (2\theta)$, $r = 5 \sin (\theta)$

A friend of mine and I have this problem for homework, and he's my math tutor for all intents/purposes. He's spent a solid hour trying to figure this out, watching videos and testing different ...
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2answers
37 views

Converting an equation from polar to recangular

I'm having trouble converting the following equation from polar to rectangular: $$ r = \frac{4}{3+2\cos(\theta)} $$ Every method I try still leaves either a $\theta$ or an $r$ in there. Can I get ...
4
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1answer
47 views

Area in Polar Coordinates

I have tried to solve this problem by subtracting the area of the whole by the smaller area. I got up to $$\int_{1/2}^{11\pi/6} 12 (\cos (\theta)-6)^2 \mathrm{d}\theta- \frac12 ...
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0answers
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$r^2\cos\theta+2ar\sin^2{\theta\over2}-a^2$ where $a>0$

What does the following equation represent? $r^2\cos\theta+2ar\sin^2{\theta\over2}-a^2$ where $a>0$ My approach: I factorized the equation and it became $(a+r\cos\theta)(a-r)=0$ I feel that ...
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0answers
30 views

area in polar coordinates

Hi! I am currently working on some calc2 online homework problems and I am having difficulty with this particular question. To be completely honest I am not sure how to even approach this problem, ...
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1answer
32 views

polar coordinates

Hi! I am currently working on some calc2 online homework problems and I am having difficulty with this problem. I was trying to use the polar coordinates (d,a)with the equation of the line thus ...
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1answer
57 views

Which conic is represented by $r = a \cos \theta$

The polar equation $r = a \cos \theta$ represents which conic?
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Drawing polar graphs when given theta in terms of the radius

I know how to plot when it is like $r=10 \cdot \sin(2\theta)$. But how to do that when the condition is like: $\theta =2 \pi \cdot \sin(r)$?
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1answer
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Polar coordinates: $ \iint_D (\sqrt{a^2 - x^2 -y^2} - \sqrt{x^2 + y^2})\:\mathrm{d}x\:\mathrm{d}y$

I need to calculate the following integral $$\iint_D \left(\sqrt{a^2 - x^2 -y^2} - \sqrt{x^2 + y^2}\right)\:\mathrm{d}x\:\mathrm{d}y$$ where $D$ is the disk $x^2 + y^2 \leq a^2$ Using the ...
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1answer
149 views

Find the area of the region inside r=a and outside r=a(1-cos θ).

I found the intersecting point to be at pi/2 and 3pi/2 a=a(1-cos θ) cos θ=0 θ= π/2, 3π/2 I'm confused as to what angles to use for integration. As for r=a, I'm assuming I'm supposed to draw a ...
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1answer
115 views

Help understanding polar coordinates and conversion between polar and rectangular

I'm not understand this. I understand that you can take normal functions with x's and y's and convert them into polar coordinates. I also understand that the polar form of that function will have the ...
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3answers
110 views

Polar Equation to Rectangular?

The equation is: $$r = \frac{4}{1+2sin(\theta)}$$ I'm confused about how to convert it into rectangular form. This is what I have so far, although I'm not sure it's correct: $$r = \frac{ ...
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0answers
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How to solve for r: rsin( θ) = sin(rcos( θ))

Please help me out. I am trying to figure out how to put y=sinx in polar form and this is as far as I can get. If this is as far as possible, then how are you supposed to graph this when you need r ...
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1answer
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Polar graph question

Can you only graph periodic functions using polar graphing? I'm not really understanding this I guess. It you are to get all of the x and y values on a finite graph, then the original must be ...
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0answers
25 views

Simplifying Inequalities Before Converting Cartesian Coords. to Polar

I have a 3 dimensional region defined by Cartesian coordinates and I have to convert them to cylindrical coordinates. That is the easy part, but what I don't understand is how to treat the ...
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0answers
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Using spirals to draw log scale?

Taking inspiration from the fact every math textbook in existence puts a picture of a nautilus shell on its logarithms chapter, I did some research, and found that many different spirals (most easily ...
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2answers
70 views

Evaluate an integral using polar $\displaystyle\int_0^2 \int_{-4\sqrt{4-x^2}}^{4\sqrt{4-x^2}}(x^2-y^2)\,dy\,dx$

How do you evaluate the following integral using polar cordinates. $$\int_0^2 \int_{-4\sqrt{4-x^2}}^{4\sqrt{4-x^2}}(x^2-y^2)\:\mathrm{d}y\:\mathrm{d}x$$ I converted it to polar coordinate making it ...
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3answers
71 views

Arc length in polar coordinates: Why isn't $dS=r×d\theta$

As a sort of exercise, I tried to derive the formula for arc length in polar coordinates, using the following logic: $$dS = r(\theta)d\theta\\ \implies S=\int r(\theta)d\theta$$ However, it turns ...
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1answer
32 views

Finding Multivariable Limits

Is there any good way to find a multivariable limit other than switching to polar coordinates? For example, students each year are inundated with problems like $$\lim_{(x,y)\to ...
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Find a vector in cartesian coordinates given its relative location to another vector in spherical coordinates

Here is my problem: -I have an arbitrary normalized vector N in cartesian coordinates -I am trying to find normalized vector M, also in cartesian coordinates -I am given the azimuth and polar ...
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1answer
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Help solving an ODE

This is an example in my book. It is for the following system: \begin{align*} x'&=y+x(1-x^2-y^2)\\ y'&=-x+y(1-x^2-y^2) \end{align*} So using polar coordinates we get the following system ...
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1answer
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How to calculate Polar coordinates for Complex Polynomials of Higher Degree?

When such I have a complex number such as $3 - 4i$, I can calculate the $r$ with $r=\sqrt{X^2+Y^2} = \sqrt{3^2+4^2}$. But how do I solve this when I have a complex number such as $(2+6i)^6$
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1answer
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Using Polar Integrals to find Volume of surface

Here's the Question and the work that I've done so far to solve it: Use polar coordinates to find the volume of the given solid. Enclosed by the hyperboloid $ −x^2 − y^2 + z^2 = 61$ and the plane $z ...
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1answer
38 views

Using polar form to show that a simple critical point is a spiral point

This is the question in my "homework." I say "homework" because it is not picked up or graded but we are supposed to do it for practice, anyhow here's the question: Given the system ...
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1answer
35 views

Changing the domain of integral

I am studying how we use polar substitution to solve double integrals. However, I am struggling with finding the correct limits of the transformed integrals to obtain a suitable solution. eg: Why ...
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0answers
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Question concerning the domain of polar coordinate.

So in the problems I encountered, I find it confusing about the domain of $\theta$. Problems take the form: For arbitrary function $f(x,y)$, and $$\displaystyle \iint_S f(x,y)dxdy=\iint_T ...
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1answer
30 views

Sinusoids closed under addition, Euler's Formula

Real sinusoids with the same frequency are closed under addition. If $$f(\omega) = A_1 \cos(\omega + \phi_1) + A_2 \cos(\omega + \phi_2)$$ Then there is some $A_3$ and $\phi_3$ so that: $$f(\omega) ...
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1answer
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For a 2 variable function, are there conditions that guarantee you can verify a limit by using only straight line trajectories?

So a recent post gave a nasty 2-variable function: $$f(x,y) = x^2y/(x^4+y^2)$$ and after changing to polar coordinates, you get that the limit is always equal to zero if you hold $\theta$ fixed and ...
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2answers
130 views

$r=5 \sec(\theta)$ into rectangular

I need to convert $r=5\sec(\theta)$ into rectangular form. I think I need to multiply both sides by $r$, because $r^2 = x^2 + y^2$, but i'm not sure how to convert $r=5\sec(\theta)$ in terms of $x$ ...
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1answer
58 views

What is the length of the cardioid $r=1-\cos(\theta)$?

I know generally how to solve this problem and was able to solve it a week or so ago. However, I keep getting stuck when trying to find $dr/d\theta$. I know that it should simplify to $-\sin(\theta)$ ...
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0answers
67 views

Shifting a plot in polar coordinates

Say we have the plot of a function $r=f(\theta)$ and want to "relocate" it to $(h,k)$. Is there a general procedure for this? I have tried the following tactic to no avail on the following example: ...
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1answer
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Trigonometry for steradian angle

Polar coordinate system is very closely associated with trigonometry. For instance, given an angle in radian, we can find its corresponding 2-D cartesian coordinates using ...
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1answer
49 views

How to get arc-length of polar function $r= 4(1-\sin{\phi})$?

How can I get arc-length of this polar function? $$ r= 4(1-\sin{\phi})$$ $$-\frac{\pi}{2}\leq\phi\leq\frac{\pi}{2}$$ I know that arc-length of polar function can get calculate by ...
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1answer
157 views

Area between two polar curves $r = 2 \sin\theta$ and $r =2\cos\theta$

I am trying to find the area between the polar curves $r = 2 \sin θ$ and $r = 2 \cos θ$. I set up the area equation as follows: $$\frac12\int_0^{\pi/4}((2\sinθ)^2-(2\cosθ)^2)\,d\theta$$ I could not ...
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2answers
51 views

Complex number polar form equation

I've been struggling with a complex numbers algebra question for a few days now, and the tutor says I still haven't got it right. Express $z_4 =−\sqrt{3} + i$ in polar form. Hence solve the ...
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1answer
32 views

Double integral And polar coordinate system

I have to evaluate this integral over the domain D The Plot would be like this: I decided to use polar coordinate system using it It gives me this but I don't know the upper limit of ...
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1answer
66 views

Problem plotting hypotrochoids using a computer

I have been trying to use a computer to plot some hypotrochoids, but I've run into some issues. For those that are unfamiliar, the parametric equations of a hypotrochoid are: $$x(\theta) = (R - ...
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204 views

Function psi, vector potential, satisfying conditions

Using spherical polar coordinates ($r, \theta, \phi$) verify that the vector $F = r^{-2}e_r$ is solenoidal. Find the function $\psi(r, \theta)$ such that $A = \frac{\psi(r, \theta)}{rsin ...
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2answers
77 views

Plotting polar equations of circles not centered at (0, 0)

Good afternoon guys! I'm fairly new to polar coordinates and polar equations, so bear with me please. I understand the equation of a circle with radius $a$ centered at the polar coordinate $(r_0, ...
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1answer
31 views

Express in polar form, $Z=0-j5$. [closed]

Express in polar form, $Z=0-j5$ $j=i$