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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1answer
110 views

find the area of the region that lies inside both curves $r=3+2\cos\theta; r=3+2\sin\theta$

find the area of the region that lies inside both curves $r=3+2\cos\theta ; r=3+2\sin\theta$ The points of intersection should be $\frac {\pi}{4} and \frac {5\pi}{4} $ I don't think these graphs are ...
1
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2answers
31 views

Is equation for ellipse in polar coordinates correct?

Wikipedia gives the following equation for the conic sections in the polar coordinate system: $r = \frac{l}{1+e\cos\varphi}$. According to the article on conic sections, in case of an ellipse $e = ...
0
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1answer
35 views

Find points near end point of a line

Any equation to find points near to both start and end points of lines with different slopes. See image. Need P and Q. If Endpoints are named A and B, AP and BQ should be 1 cm
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0answers
16 views

Heat equation on a circular plate

I'm in trouble with the following problem: assuming a circular plate of radius $R$, the heat equation on it, is: $$\partial_t ...
0
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1answer
108 views

graph polar coordinates $ r=4\sin(3\theta) $

Graph polar coordinates $ r=4\sin(3\theta) $ I was told by my teacher to split the graph into $3$ parts per quadrant and try those angles the problem arises when I plug $\dfrac{5\pi}{6}$ into the ...
0
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1answer
29 views

Tranforming to polar co-ordinates

$$I = \int_0^1\int_0^{\sqrt{1-x^2}} xy \, dy\, dx$$ By transforming to circular polar co-ordinates, evaluate I. How do I do this? Is there a formula/strategy for doing this that works with ...
0
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1answer
41 views

Convert $r^2\cos(2\theta)=9$ to Cartesian

I need to convert $r^2\cos(2\theta)=9$ to Cartesian coordinates. How should I do it? What I did: $$r^{2}\cos2\theta=r^{2}2\cos^{2}\theta-1=9\Rightarrow r^{2}\cos^{2}\theta=5\Rightarrow x^{2}=5$$ Did ...
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2answers
36 views

Cartesian coordinates of lemniscate

A problem from Spivak's Calculus: Sketch the graph of the lemniscate $$r^2 = 2a^2\cos 2 \theta.$$ Find an equation for its cartesian coordinates. Show that it is the collection of ...
2
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1answer
39 views

Finite differences to ODE in polar coordinates

I have an equation of finite differences as follows: $$\frac{X_1(r+\epsilon)-X_1(r)}{ \frac{\epsilon~\beta}{2~r+\epsilon} } + \frac{X_1(r-\epsilon)-X_1(r)}{ \frac{\epsilon~\beta}{2~r-\epsilon} } = ...
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1answer
16 views

Center of Mass double Integral using polar Coord.

Find center of mass given Lamina pictured: https://s3.amazonaws.com/wamapdata/qimages/qtrring.gif with inner radius of 3 and an outer radius of 7, and a density function $$\rho(x,y) = ...
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2answers
31 views

Surface: intersection of 2 polar curves

I have these two polar curves: $$ C_1: r = 2 - \cos(\theta)\\ C_2: r = 3 \cos(\theta) $$ Plots: C1 and C2. I need to find the surface of $D = C_1 \cap C_2$. I started by finding the solution to ...
0
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2answers
41 views

how do we interpret this integral from polar co-ordinates

$$\text{Find } \int_C rdr$$ Where $C$ is any closed loop. I feel that the answer is zero, i have no hard reasoning. Here $r$ is the parameter from the polar coordinates.
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0answers
32 views

Double integral polar coordinate using substitution

I have to calculate: $$\iint _{R} \frac{x}{\sqrt{x^{2}+y^{2}}}dA$$ and R is this region: $$x^{2}+y^{2}=16; x^{2}+y^{2}=4; y = \sqrt{3}x; y=\frac{x}{\sqrt{3}}$$ so I used substitution: $$x = r\cos ...
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1answer
26 views

Find minimum of Archimedean spiral in revolution

I just recently asked about limacon curves, but now I want to examine the Archimedian curve: $$r=a + b\theta$$ The question is essentially the same, for a given range, how would one find the minimum ...
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4answers
40 views

Find minimum x value from a polar function

I am mainly examining limacon functions. For the equation r= b + a*cos(theta), it is easy to find the minimum radius, but I want to find the most negative value (between a given range). Take function ...
4
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1answer
151 views

Multiple integral over a disc

I would need some help on this integration problem: $$I=\int_0^{2\pi}\int_0^{R}\int_0^{2\pi}\int_0^{R}\exp(-a\ r_{12}) \ r_1 \ r_2 ...
2
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1answer
22 views

Why is the integral of the arc length in polar form not similar to the length of the arc of a circular sector?

So I learned that the area enclosed by a polar function is computed by $$A = \int \frac{r(\theta)^2}{2}d\theta.$$ Which, I learned, comes somewhat from the formula for the area of a circular sector ...
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1answer
36 views

Ellipse region in polar coordinates

if I want to write the region in $R^2$ bounded by the ellipse $$10x^2 + 17 y^2 = 29$$ In polar coordinates($x=r\cos \theta, y= r \sin \theta$), how can I find the limit of $r$?
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1answer
32 views

Calculate if a Circle intersects a Arc

Have a Cartesian Plane cartesian plane And a Arc with the measures: point = 200, 200 radius = 50 start angle = 0 end angle = 180 And a Circle with the ...
3
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1answer
66 views

approximate this fancy looking double integral

$$\int_{0}^{2\pi} \int_{0}^{1}r^5\sin^22\theta\left(1-r^2 \right)^2\sqrt{1+\left(1+ \cos^2\theta \right)36r^2 }\hspace{1mm}drd\theta$$ I tried integrating myself, spent many hours but could not ...
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2answers
35 views

Polar coordinates in the cartesian plane.

${dy}/{dx} = {dy}/{d\theta}$ divided by $dx/d\theta$ where $x$ and $y$ are in the Cartesian plane and $\theta$ is in the polar plane and $x = r\cos( \theta), \ y = r \sin (\theta)$. If $dy/dx = 0$ ...
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2answers
104 views

converting kph and heading to xyz velocity vector

I am writing software (in C++) that is required to send out messages from our simulation system to another simulation system. Problem is we track the simulation object's current speed (kph) and ...
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1answer
26 views

Area in a polar curve question

C1: $\;r= 3 \sin x$ where $0\leq x\leq \pi$ C2: $\;r= 1 + \sin x$ where $-\pi \leq x \leq \pi$ Please help me in this question I have drawn the sketch of the two polar equations. Then the ...
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1answer
32 views

Computing the enclosed area formed by a curve

Given the following curve: ${(\frac{x^2}{a^2}+\frac{y^2}{b^2})}^2=\frac{x^2}{a^2}-\frac{y^2}{b^2}$, $a,b>0$ Find the space enclosed by it. Now, obviously (I think) the idea is to ...
2
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2answers
91 views

Calculate area of $r^2 = \cos(2 \theta)$ without breaking into individual petals?

In the area integral, I am integrating first $r$ from: $-sqrt\cos(2\theta)$ to $sqrt\cos(2\theta)$ and theta from $-\pi/4$ to $\pi/4$. Integrand is $r*dr*d\theta$. In r integral it comes 0 after ...
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1answer
58 views

how to find slope of this polar curve: $r^2=\sin(2\theta)$.

Given $r^2=\sin(2\theta),\;$ how to find the slope of the tangent line at $x=0$ ? If the question were $r=\sin(2\theta)$, it would be o.k. but since it is $r^2=\sin(2\theta)$, I don't know how to ...
0
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1answer
78 views

Find k in $\int_2^{\infty} \frac{k}{\sqrt{2\pi}} \exp^{-\frac{1}{2} x^2} \, dx$

I'm trying to solve for k in the pdf: \begin{equation} \int_2^{\infty} \frac{k}{\sqrt{2\pi}} \exp^{-\frac{1}{2} x^2} \, dx \end{equation} My solution (which is wrong): Take the square of the ...
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1answer
36 views

Integral formula for polar coordinates

The polar coordinates of point $x \in \mathbb{R} \setminus \{0\}$ are pairs $(r,\gamma)$, where $0 < r < \infty$ and $\gamma \in S^{d-1} = \{x \in \mathbb{R}^{d}\mid |x| = 1\}$. These are ...
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0answers
31 views

Differential Operators in different coordinates

How does one show this identity? $$\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}=\frac{\partial^2}{\partial r^2}+{1\over r}\frac{\partial}{\partial r}+{1\over ...
0
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1answer
27 views

If $f(2\alpha-\theta) = f(\theta)$, then $\theta=\alpha$ is a line of symmetry of $r=f(\theta)$. How do you derive $f(2\alpha-\theta) = f(\theta)$?

For Polar Coordinates I know that for x-axis symmetry $f(-\theta)=f(\theta)$, for y-axis symmetry $f(\theta)=f(\pi-\theta)$, and for symmetry about the origin $f(\theta)=f(\theta+\pi)$. The big ...
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0answers
28 views

polar co ordinates integration

integrate the polar co ordinates $$ \int^{r=\infty}_{r=0} \int^{z=\infty}_{z=-\infty} \delta(r) \delta(z-z_s) dz dr$$ => I want to integrate the above equation. integral of $ \int^ {z=\infty} _ {z= ...
0
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1answer
43 views

Volume of solid bounded by $z^2 = x^2 + y^2$ and $x^2 + y^2 = 2x$

Calculate the volume of the solid bounded by $z^2 = x^2 + y^2 $ and $x^2 + y^2 = 2x$ My attempt: Using cylindrical coordinates, $$ \mathrm{Vol} = \int_{-\pi/2}^{\pi/2} \int_0^1 \int_{-r}^{r} r ...
2
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1answer
55 views

Cylindrical cordinates: $\iiint (x^2 + y^2 + z^2) dxdydz$

Show that $$ I= \iiint_S (x^2 + y^2 + z^2) dxdydz = \frac{2^{10} a^5 k}{75} \left(1 + \frac{k^2}{3} \right), a>0, k>0$$ where $S$ is the region bounded by the cilinder $x^2 + y^2 = 2ax$ and ...
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1answer
35 views

Convert the polar equation to Cartesian coordinates : $r^3 = − 7cos\theta$

I have a question to convert $r^3 = − 7cos\theta$ into cartesian coordinates. I'm having a hard time understanding what to do. I'm familiar with converting a polar coordinate to a Cartesian ...
0
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1answer
49 views

Arc Length polar curve

$$r=a\sin^3\left(\frac{\theta}{3}\right) $$ I tried solving it using the equation for arc length with $dr/d\theta$ and $r^2$. Comes out messy and complicated.
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2answers
33 views

Plotting complicated polar curve without calculator

I'm wondering if anyone can give me tips or guidance on how to plot complicated polar curves without the use of a calculator. Most notably, I am trying to plot: Based on a graphing calculator, I ...
2
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1answer
36 views

Representation of cardiod in the complex plane

I noticed that the complex function $$f(z) = \frac{2}{(z+i)^2}$$ seems to map the real line onto the cardioid given by the polar equation: $$r = 1- \cos(\theta).$$ I was wondering if there is a simple ...
0
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0answers
28 views

Transformation in polar coordinate system

I have a point $P \in \mathbb{R}^2$. $P$ in a local Cartesian coordinate is given as $(x,y)$ or alternatively the polar representation is given as $(\rho, \theta)$. This local coordinate is located ...
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0answers
21 views

Polar coordinates: Slope of tangent

Would anyone mind telling me how to solve this problem? It seems strange as my answer is $-1$. Do I have to apply this formula, $(r'sinθ +rcosθ)/(r'cosθ -rsinθ)$ ?
2
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1answer
69 views

Converting Cartesian circle to polar form

I am trying to convert circle equation from Cartesian to polar coordinates. I know the solution is all over the Internet but what I am looking for is the exact procedure and explanation, not just the ...
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1answer
55 views

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
123 views

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: ...
7
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3answers
87 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
51 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 ...
2
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1answer
78 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
30 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
62 views

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
39 views

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
40 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? $$ ...
0
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1answer
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 ...