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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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
607 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
89 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 ...
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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
41 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
49 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
38 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
29 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
327 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
34 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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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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0answers
19 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
48 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
8 views

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
67 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
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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
29 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
31 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
19 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 ...
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1answer
77 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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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
25 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) = ...
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2answers
21 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
40 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} ...
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43 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
118 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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0answers
51 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
63 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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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
42 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 ...
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1answer
45 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
35 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 $$ ...
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1answer
39 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 ...
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1answer
91 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 ...
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39 views

Dot product of gradient and tangent vector

Using polar coordinates with variables $r$ and $\theta$. Let $\vec{r}$ be the position vector. Consider $\nabla \theta \cdot \frac{d\vec{r}}{d\theta}$. This is the dot product of the gradient normal ...
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12 views

Polar Coordinates to Cartesian - Finding Y component

I have the following diagram and frame: I am trying to find out what the equation is that matches XYZ to RThetaPhi. Basically, I need an expression that gives Ys in terms of RThetaPhi. My problem ...
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58 views

Difference between Euler form and polar / trig form of a complex number

After some readings, I have found out that the difference between the polar / trigonometric form and the Euler form of a complex number consists on the fact that in the first case is expressed the ...
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1answer
26 views

Double integrals in polar coordinates — Multivariable

I've done some research on this topic but I am quite confused about finding the area under a specific volume in polar coordinates. Let's have an example, how would we find the volume of a hyperboloid ...
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2answers
80 views

find the area of the region lying inside the circle $r=6$ and inside the cardioid $r=4-3\sin \theta$.

Well, I drew a graph to visualise it and I found the interceptions $\theta=\arcsin \left(-\frac{2}{3}\right)$. From the graph, by symmetry, I found that the area of region from $\theta$ to $\pi/2$ and ...
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1answer
130 views

Find the area of the region lying outside a circle r=7 and inside the cardioid r=6+7sin theta

So this is the question I have problem dealing with. I know that firstly I need to equate $7$ and $6 + 7\sin \theta$ to get the intersection. And then I am supposed to apply the formula.. But I am ...
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0answers
23 views

Arc Lengths of Points Tangent to a Logarithmic Spiral

Suppose we are given distinct array of $N$ vertices (or Cartesian points) $V_n =(v_1, v_2, ... v_n), v_i \in \mathbb{R}^2$. Taking $v_1$ to be the origin of a logarithmic spiral whose curve ...
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25 views

Predict a point when you are given initial measurements

From given (x,y) sensor measurements, output by a robot, I need to find robot's heading direction and predict the next location. I have an algorithm that when programmed gives me the correct answer, ...
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0answers
62 views

Tangent of angle between tan line and radial line

How can I use the fact that if the curve whose polar equation is $r=f(\theta)$ is rotated about the pole through an angle $\phi$, then an equation for the rotated curve is $r=f(\theta-\phi)$ to prove ...
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0answers
10 views

Scale factors and metric in cylindrical and spherical coordinates - isotropy of space [duplicate]

In cylindrical (polar) coordinates, the scale factors are $$h_r=1$$ $$h_{\theta}=r$$ $$h_z=1$$ Would it be correct to say that $h_i$ do not depend on $\theta$ because space is isotropic (has the same ...
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0answers
76 views

Scale factors in cylindrical coordinates - geometrical meaning

I am trying to make sense of the scale factors in cylindrical coordinates and their geometrical meaning. To start with something simpler, begin with Cartesian coordinates: $$h_x=h_y=h_z=1$$ One can ...