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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16 views

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
396 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
40 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
50 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
36 views

$r^2\cos\theta+2ar\sin^2{\theta\over2}-a^2$ 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
31 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
34 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
66 views

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

The polar equation $r = a \cos \theta$ represents which conic?
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1answer
78 views

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
47 views

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
272 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
161 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 ...
2
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3answers
140 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
35 views

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
39 views

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
26 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
19 views

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
78 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 ...
2
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3answers
104 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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2answers
42 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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1answer
22 views

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 ...
3
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1answer
29 views

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
32 views

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
32 views

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
42 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
36 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
31 views

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 ...
2
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1answer
34 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
16 views

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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5answers
1k views

Limit is found using polar coordinates but it is not supposed to exist.

Consider the following 2-variable function: $$f(x,y) = \frac{x^2y}{x^4+y^2}$$ I would like to find the limit of this function as $(x,y) \rightarrow (0,0)$. I used polar coordinates instead of ...
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2answers
160 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
66 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 $-\sin(\theta)$ ...
2
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0answers
98 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
62 views

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
60 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 ...
3
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1answer
234 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
61 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
35 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
77 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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0answers
224 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
97 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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0answers
59 views

Algorithm for finding nearest distance from a point to a curved surface in space

I need to write an algorithm which can find the nearest distance from a point in space to a 3D curved surface which is straight in vertical direction but its projection is an arc of a circle (Similar ...
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1answer
103 views

Express this polar equation in cartesian form

Having trouble converting this polar equation into Cartesian form: $r = 2 + \sin(\theta)$ This is how far I get: $(r = 2 + \sin(\theta))\cdot r$ $r^2 = 2r + r\sin(\theta)$ $x^2 + y^2 = 2r + y$, ...
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0answers
62 views

Finding an arc-length between 2 points in 3 dimensions

I know how to find an arc-length between two points with coordinates, say $X=(a,b)$ and $Y=(c,d)$. But how do I find the same thing but for, say $X=(a,b,c)$ and $Y=(d,e,f)$? Thanks!
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0answers
121 views

Wrong answer within 'Calculus Solution Manual, Michael Spivak, 3rd ed'

I have a problem with the answer provided in the solution manual of Calculus, Michael Spivak, 3rd ed, The Problem: Consider a hyperbola, where the difference of the distance between the two foci is ...
2
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4answers
283 views

Linear combinations of sine and cosine

If you take a linear combination of the cosine and sine function, then the result is again a sinusoid, but with a new amplitude and phase shift. $$a \cos(\theta) + b \sin(\theta) = A \cos(\theta + ...
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1answer
75 views

Polar equation — find area under graph using double integral

What is the area of the region in the plane bounded by the curve given in polar coordinates $r = 4 + 2\cos(2\theta)$? Could someone walk me through the conversion of polar coordinates to rectangular ...
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1answer
71 views

What is this called: $ \frac{\partial^2f}{\partial x^2} + \frac{\partial^2f}{\partial y^2} = $ … Laplacian?

$ \frac{\partial^2f}{\partial x^2} + \frac{\partial^2f}{\partial y^2} = \left( \frac{\partial^2 f}{\partial r^2} + \frac{1}{r^2}\frac{\partial^2 f}{\partial \theta^2} + ...
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3answers
61 views

Is the graph of $r^2 = 4$ a circle with radius $2$?

If $r^2 = 4$, taking the square root of both sides will give me $r = 2$, so its graph is a circle with radius $2$. Is this correct? I just wanted to make sure because $r^2$ might imply another graph.
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3answers
66 views

What is the equivalent polar equation of $x^2 + (y-1)^2 = 1$?

It's a question in the textbook that I have and I am having a hard time understanding it. How am I supposed to get the polar equation with this format?