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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51 views

Non-simultaneous intersections of $r = 4\cos\theta+1$ and $r = 2\cos\theta+1$

$$r = 4\cos\theta+1$$ $$r = 2\cos\theta+1$$ This system has simultaneous solutions at $(1, \frac\pi2)$ and $(1, \frac{3\pi}2)$. But looking at the graph, there are non-simultaneous intersections at $...
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1answer
76 views

Applied Mathematics: Spherical Polar Coordinates and Newton's Second Law

I've been attempting this question but can't seem to find a solution. Question: A particle of mass $m$ moves under the influence of a force which, in spherical polar coordinates, only acts in the ...
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1answer
50 views

Find the points on the given curve where the tangent line is horizontal or vertical.

Please help! I don't know how else to do this question. Thank you!!
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1answer
21 views

Max and minimum value that function $x*e^{x^2+y^2}$ can take on D

So I have to find the maximum and minimum value that the function $~xe^{x^2+y^2}~$ can take on: $$ D = \bigl\{(x,y) :\, 9 \leq x^2 + y^2 \le 16,~ y \geq 0\bigr\} $$ I've converted the Cartesian ...
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2answers
25 views

Exponential to polar form

I have exponential form $$ je^{-j\pi/2} $$, where $j = \sqrt{-1}$ I want to convert this to polar form $$j(\cos\pi/2 + j \sin \pi/2)$$ is it correct?
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1answer
35 views

Polar coordinates: what is the area of the region inside the inner loop of $r = \cos (\theta) - \frac12$?

I'm struggling plotting $r = \cos (\theta) - \frac12$. I've done it in Cartesian but I can't quite get in polar coordinates. I know it is supposed to be a loop but how do I get it? Being that I have ...
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1answer
26 views

Change of variables - Double integrals

I have trouble understanding how the limits work regarding polar coordinates in a double integral. For example, say if I had the equation $$(x-2)^2 + y^2 = 1.$$ This is a circle centred at (2,0) with ...
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1answer
79 views

Evaluate the double integral by changing to polar coordinates

I experience some difficulty with converting to polar coordinates in integrals. So the question I'm struggling with is Evaluate the double integral $$\int\int x^{6}y\, dA$$ where $D$ is the top ...
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1answer
12 views

What PC programs or iPad applications are there which allow you to plot cylindrical/spherical polar graphs?

I've been trying to get my head around the use of cylindrical and spherical polars to plot graphs. I feel that the easiest way to do this would be to try plotting some, but I'm struggling to find a ...
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1answer
15 views

How would you use cylindrical polar coordinates to find the area of a cone (and why does my method not work?

The following question was recently asked in a lecture: Using cylindrical polar coordinates find the area of the curved surface of a cone of height $h$ and radius $a$. My attempt to do this was ...
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0answers
507 views

How to solve this integral to find the exact length of an equation in the polar plane?

I hope it is only because it's late and I've been studying for a calculus exam for several hours, but I cannot see how to solve this integral. The problem states: Find the exact length of the ...
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1answer
29 views

Complex Numbers Midpoint of Roots of Unity

A = $\sqrt{2}e^{i(\frac{7\pi}{12})}$ B = $\sqrt{2}e^{i(\frac{11\pi}{12})}$ Express the midpoint M of AB in the form $a + bi$ (a,b in simplified surd form) I know M = (A+B)/2 but I cant find A+B in ...
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2answers
866 views

Find the area of the region that lies inside the first curve and outside the second curve. $r = 10 \cos\theta,\ r = 5$

I am not sure of my answer. In the figure, $r=10 \cos\theta$ is a circle that doesn't look like a circle. The area of $r=5$ is $\pi r^2 = 25 \pi$. You remove the area from $-\pi/3$ to $\pi/3$ of $...
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2answers
90 views

How to get the area between these $2$ functions?

I have a function: $(a)$ $r = 4\cos(2\theta)$ $(b)$ $r = 4\sin(2\theta)$. I need at least a set up for the integral that will yield the area inside the rose (a) but outside the rose $(b).$ I cant ...
71
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3answers
1k views

Cardioid in coffee mug?

I've been learning about polar curves in my Calc class and the other day I saw this suspiciously $r=1-\cos \theta$ looking thing in my coffee cup (well actually $r=1-\sin \theta$ if we're being ...
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4answers
42 views

Does an integral of a polar function from $0$ to infinity have to diverge?

This is more of a theoretical question, but I was curious if a polar equation automatically diverges as it goes to infinity? After all, the area will just be the area in the polar graph added to ...
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3answers
53 views

Evaluate iterated integral by changing to polar coordinates

$$\int_0^{1/2}\int_0^{\sqrt{1-x^2}}xy\sqrt{x^2+y^2}\,dy\,dx$$ $x^2+y^2=r^2$ $$\int\int_0 r^3\cos\theta \sin\theta|r|\,dr\,d\theta$$ I don't know what $r =$ at line $x = 1/2$. I don't know value of $\...
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3answers
39 views

Converting Polar Equation to Cartesian Equation problem

So I have 1. $$\frac{r}{3\tan \theta} = \sin \theta$$ 2. $$r=3\cos \theta$$ What would be the Cartesian equation???
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4answers
31 views

Polar Equation to Cartesian Equation.

Find the cartesian equation of the circle with polar equation $r=2a\cos \theta$ My attempt, Since $\cos \theta=\frac{x}{r}$ So, $r=2a(\frac{x}{r})$ I don't know how to proceed.
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2answers
371 views

Spiral of Archimedes area and sketch in polar coordinates

This is an exercise from Apostol's Calculus, Volume 1. It asks us to sketch the graph in polar coordinates and find the area of the radial set for the function: $$f(\theta) = \theta$$ On the interva ...
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0answers
37 views

Rewrite the equation of a conic in cartesian coordinates

Consider the equation for a conic in polar co-ordinates $(r,\theta)$ $$r = \frac{k}{1 - e\cos(\theta)} \qquad \qquad (1)$$ in the case where $k > 0$ and $e > 1$. Show that ...
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0answers
15 views

How would you express this integral in cylindrical polar coordinates?

How would you express the integral \begin{gather*} \int_{0}^{1}\int _{0}^{\sqrt{1-x^{2}}}\int_{0}^{1-x^{2}-y^{2}} e^{z} \ dz \ dy \ dx \end{gather*} In cylindrical polar coordinates, would it be ...
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23 views

How to read a 3D Polar Plot?

r = 1 - sin[n*theta] I'm an integral calculus student and I'm trying to interpret 3D polar graphs. I know that r is the amplitude for polar coordinates and n is a scalar for theta which dictates the ...
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3answers
50 views

The value of the cubic root of $-i$

So this was the question given to us. $\left(\iota=\sqrt{-1}\right)$ Value(s) of $\left(-\iota\right)^{\dfrac{1}{3}}$ are (A) $\dfrac{\sqrt{3}-\iota}{2}$ (B) $\dfrac{\sqrt{3}+\iota}{2}$ ...
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0answers
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Partially differentiating interrelated equations

I apologize if this question is too simple or if I am not following guidelines in any way. This is my first question, so constructive criticism is appreciated. Anyway, I am given Z = x^2 +2y^2 x = ...
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1answer
18 views

Finding the polar form of a complex number

I have the following complex numbers : -3,18 +4,19i I can calculate $r=\sqrt{a^2+b^2}$ Which gives r=5,26 now I know that cos $\theta = \frac{a}{r}$ gives $\theta=127,20$ degrees But when I do ...
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2answers
77 views

Double integral with Polar coordinates - hard example

Calculate using polar coordinates: $$\iint_{D}^{} (x^2+y^2)^\frac{1}5 \ dx \ dy $$ where D is the region inside the circle with radius 1. Working: D: $ \ x^2+y^2=1 \\ $ so $ 0 \leq r \leq 1 \ \ , $ ...
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0answers
34 views

Operators in polar coordinates in n-dimensions

I want help on converting differential operators such as the reduced wave operator (L=Δ+c) and the biharmonic operator (L=Δ^2) from Cartesian to spherical coordinates in n-dimensions. For example I ...
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1answer
30 views

Finding the area of a polar region

I am trying to find the area inside the curve $$ r = 2 + \sin2\Theta + \cos3\Theta .$$ It's a very weird looking function after graphing, and I'm not quite sure how I'm supposed to proceed. There's ...
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1answer
33 views

Parametrization of a rotating surface

What is the parametrization of a surface obtained by rotating the circle $(y − 3)^2 + z^2 = 1, x = 0$ about the z-axis. I came up with the parametrization $S(r,θ) = (r , 3+cosθ , sinθ)$, is it ...
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2answers
23 views

How to calculate polar angle of point given a reference point?

I want to calculate polar angle of some points based on different reference points. Usually polar angle is calculated based on reference point (0,0). What is the procedure to calculate polar angle ...
6
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1answer
105 views

Simple proof of the Cauchy-Crofton formula on the sphere?

Let $\gamma$ be a regular curve on the sphere. In a lecture, the following result was used $$L(\gamma)=\frac 14 \int_{S^2} \sharp (\gamma \cap \xi ^\perp)d\xi$$ $\xi^\perp$ is the plane with normal $...
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0answers
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Does a plane curve with polar equation $r=\lambda_1\cos^2\theta+\lambda_2\sin^2\theta$ have a name?

Does a plane curve with polar equation $$r=\lambda_1\cos^2\theta+\lambda_2\sin^2\theta$$ where both $\lambda_i>0$ have a name? It's very similar to hippopede, also known as lemniscate of Booth, ...
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1answer
28 views

Integral that results in the fraction of two gamma functions

I'm trying to show this equation $$ \int\limits_0^\infty \mathrm{d}x_1 \dots \mathrm{d}x_n \left( 1 - \sum_{i=1}^n x_i \right)^k \Theta \left( 1 - \sum_{i=1}^n x_i \right) = \frac{\Gamma(k+1)}{\Gamma(...
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2answers
23 views

Parametric Equation of conics: Parabola

Let $P(ap^2,2ap)$ and $Q(aq^2,2aq)$ be two points on the parabola $y^2=4ax$ such that PQ is the focal chord. Let $A(at^2,2at)$ and $B(as^2,2as)$ be two other variable points on $y^2=4ax$. a) Show ...
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1answer
41 views

Double checking a change of variables to Polar in a differential equation

I have $$\frac {\mathrm{d}x}{\mathrm{d}t}=x-y-(2x^2+y^2)x $$$$\frac{\mathrm{d}y}{\mathrm{d}t}=x+y-(x^2+2y^2)y$$ I have calculated $\frac{\mathrm{d}r}{\mathrm{d}\theta} =r+r^3(\frac12\sin^2(2\theta)-...
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44 views

Angle using cross product

I have a situation. Please refer to a figure below: I have r1, r2, Ɵ1, Ɵ2 as well the reference line. I want to find out angle Φ(phi). i.e. (angle PBA). Edit_1 The link provided solves the ...
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2answers
144 views

Find the volume of the solid bounded above by the cone $z^2 = x^2 + y^2$, below by the $xy$ plane, and on the sides by the cylinder$ x^2 + y^2 = 6x$.

Q: Find the volume of the solid bounded above by the cone $z^2 = x^2 + y^2$, below by the xy plane, and on the sides by the cylinder $x^2 + y^2 = 6x$. I can't figure out what I'm doing wrong in my ...
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53 views

How is double integral variable substitution different from one variable trigonometric substitution?

I'm studying variable change in double integrals and I understood the reasoning behind the formulas as described really well here. However, geometric arguments for analysis don't convince, as well as ...
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1answer
30 views

Convert Cartesian function to polar function

A problem on my math homework is $x = -4$ convert to a polar function. What are the steps, the examples in my book are only for $y=x$ functions.
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1answer
78 views

Plotting a exponential form of complex number over an angle on an Argand Diagram

Say I had to plot the expression $$\frac{\pi e^{i\theta }}{4\theta}$$ where $\frac{\pi}{4} \le \theta \le \frac{9\pi}{4} $ on an Argand diagram, how would one go about doing so? If it was just the ...
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17 views

Domain in polar coordinates with a square and a discus

I was doing some studying in Steward's Calculus when I came onto this problem. I am asked to integrate a certain function $f(x,y)$ in this domain. I know how to do it when the inner boundary is a ...
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0answers
43 views

Polar coordinates for double integral for $\theta$

To evaluate the integral $$\iint_D \sqrt{x^2+y^2}dA$$ $$D=\{(x,y)\mid0\leq(x-1)^2+y^2\leq1 \}$$ it should be best to change variables into polar coordinates to get $$\int_{\frac{-\pi}{2}}^{\frac{\pi}...
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1answer
46 views

Calculate surface area of flat figure by using double integral and polar coordinates

Check me please. I tried check it via WolframAlpha, but I don't trust in it 100%. Task: Calculate surface area of flat figure by using double integral in polar coordinates. Figure confined by line: ...
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Derivation Check: Point Described by Cylindrical Coordinates to Euler Angles

this post is quite long so thank you in advance for those who get through it. I made this post as a confirmation of my logic so others can check over my work and since I didn't find any posts relating ...
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2answers
36 views

Calculate double integral using Polar coordinate system

Need to calculate $\int_{0}^{R}dx\int_{-\sqrt{{R}^{2}-{x}^{2}}}^{\sqrt{{R}^{2}-{x}^{2}}}cos({x}^{2}+{y}^{2})dy$ My steps: Domain of integration is the circle with center (0,0) and radius R; $x = \...
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0answers
25 views

Evaluating $\iiint_R \log\Big((x^2 + y^2 + z^2)^\frac{3}{2}\Big)\, dx\ dy\ dz$ between balls in $\Bbb R^3$

I am working on the following problem: Evaluate: $$\iiint_R \log\Big((x^2 + y^2 + z^2)^\frac{3}{2}\Big)\, dx\ dy\ dz,$$ where $R = \big\{(x, y, z) : 1 \leq x^2 + y^2 + z^2 \leq 2^2 \big\}$ is the ...
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1answer
36 views

Relating $dS$ and $d\theta$ for computation of line integrals

I'm asked to compute $$\int_C \vec{F} \cdot d\vec{s}$$ where $\vec{F} = A_0(x\hat{y} - y\hat{x})$, along a circular path, counterclockwise, about the origin with radius 4. I begin by writing $$ \vec{...
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1answer
42 views

What are the characteristics of functions that look the same in both polar and rectangular graph?

Today, I am doing practice for SAT. In a textbook example, I see $$r=\frac{1}{\sin\theta}$$ My textbook is telling me that this particular function looks the same whether it's graphed on a polar or ...
2
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1answer
116 views

Double integration in polar coordinates between two circles

I am trying to integrate converting to polar coordinates, between two circles. $$A = \iint_D x \,\mathrm{d}A $$ Ant the domain of integration is set to be the region in the first quadrant between ...