1
vote
2answers
28 views

Evaluating a polar double integral on the semi disc

The integral: $$\iint_D (x^2-y^2)\,dx\,dy$$ where $D$ is defined as: $$\{(x,y)\in \mathbb R^2 \mid x^2+y^2\le 1, x\ge 0\}$$ Context I have solved double integrals on quarter discs but this semi ...
1
vote
1answer
31 views

Area of a Self-intersecting Curve

I was doing some work finding the areas of rose curves. The rose curve is a polar curve given by the equation $$ r(\theta) = \cos{k\theta} $$ When $k$ is even, the area is $\pi/2$, and when $k$ is ...
3
votes
0answers
62 views

My orbiting body is orbiting about the wrong focus of it's elliptical orbit… why? [closed]

I am coding in c++ and am computing the position of an orbiting body as a function of time. Everything is almost working. I have a nice elliptical orbit. Except, my orbiting body speeds up as it ...
1
vote
1answer
62 views

Finding a mistake in the computation of a double integral in polar coordinates

I have to find $P\left(4\left(x-45\right)^2+100\left(y-20\right)^2\leq 2 \right) $ $f(x)$ and $f(y)$ are given, which I will use in my solution below . ...
3
votes
3answers
288 views

Evaluation of the integral of $e^{-(x^2+y^2)}$ over a disk

Show that $$\renewcommand{\intd}{\,\mathrm{d}} \iint_{D(R)} e^{-(x^2+y^2)} \intd x \intd y = \pi \left(1 - e^{-R^2}\right)$$ where $D(R)$ is the disc of radius $R$ with center $(0,0).$ I ...
3
votes
1answer
46 views

Use a double integral in polar coordinates to find the area

So the area is just an intersection of two circles Converting the two circles to polar coordinates, I get: $r(r-2\sin\theta)=0$, and $r(r-2\cos\theta)=0$ Ummm so $r =0$ and r = $2\sin\theta$ ...
1
vote
1answer
46 views

How to prove that the graph of $r=\sin(\frac{\theta}{2})$ is symmetry about polar axis

I want to know how to prove that the graph of $r=\sin(\frac{\theta}{2})$ is symmetry about the $x$-axis(polar axis). As I understand, if a polar graph is symmetrical about $x$-axis, $(r,\theta)$ and ...
0
votes
2answers
30 views

Find the area of the circle

Find the area of the circle defined by the parametric equations $x = \cos t$ and $y = \sin t$. I know this is circle defined by $x^2 +y^2 =1$ so i used $0 < t < 2\pi$ as my bounds, then ...
2
votes
2answers
64 views

Integrating $ \int_0^2 \int_0^ \sqrt{1-(x-1)^2} \frac{x+y}{x^2+y^2} dy\,dx$ in polar coordinates

I'm having a problem integrating $ \displaystyle\int_0^2 \int_0^ \sqrt{1-(x-1)^2} \frac{x+y}{x^2+y^2} \,dy\,dx$. I drew the graph, and it looks like half a circle on top of the $x$ axis. I tried ...
0
votes
3answers
55 views

Integrating $\int_1^2 \int_0^ \sqrt{2x-x^2} \frac{1}{((x^2+y^2)^2} dydx $ in polar coordinates

I'm having a problem converting $\int_1^2 \int_0^ \sqrt{2x-x^2} \frac{1}{(x^2+y^2)^2} dy dx $ to polar coordinates. I drew the graph using my calculator, which looked like half a circle on the x ...
2
votes
3answers
45 views

graph the curve and find its length, $r=\cos^2(\frac {\theta}{2}) $

graph the curve and find it's length, $r=\cos^2(\frac {\theta}{2}) $ I graphed it and found that it was a cardioid (or a sideways heart). I am getting stuck on the arc length. this is what I have: ...
-1
votes
1answer
72 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 ...
0
votes
1answer
83 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
votes
1answer
35 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 ...
0
votes
2answers
39 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.
3
votes
1answer
64 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 ...
0
votes
2answers
33 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 $dy/dx = 0$ ...
1
vote
1answer
48 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
votes
1answer
76 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 ...
0
votes
1answer
42 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.
5
votes
5answers
119 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: ...
0
votes
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 ...
0
votes
1answer
278 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 ...
4
votes
1answer
49 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 ...
1
vote
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, ...
0
votes
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 ...
3
votes
1answer
46 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 ...
1
vote
1answer
214 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 ...
0
votes
1answer
31 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 $(2+6i)^6$
0
votes
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 ...
1
vote
0answers
26 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
votes
0answers
85 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: ...
0
votes
1answer
54 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
votes
1answer
193 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 ...
1
vote
1answer
90 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: $(r = 2 + \sin(\theta))\cdot r$ $r^2 = 2r + r\sin(\theta)$ $x^2 + y^2 = 2r + y$, ...
2
votes
0answers
91 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
votes
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.
1
vote
3answers
61 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?
1
vote
1answer
193 views

What is the cartesian equation of $r = 1 + r \sin(\theta)?$

There are no values given for $r$, or $\theta$. How do I derive the cartesian equation for this? It's a question from a textbook I have.
0
votes
1answer
30 views

A polar integration question

I'm trying to prove this integral $$ \int_0^a \int_0^\sqrt{a^2-x^2} f(x,y) \, \mathrm{d}y \, \mathrm{d}x$$ is the same as $$\int_0^{2\pi} \int_0^a r f(r,\theta) \, \mathrm{d}r \, \mathrm{d}\theta$$ I ...
1
vote
1answer
57 views

Using integration and polar coordinates to find the volume of a torus

How would I find the volume of the body formed by revolving the circle $r = f(\theta) = \cos\theta$ about the line $\theta = \frac{\pi}{2}$ ? (This is the circle of radius $1$ centered at $(0,1)$ ...
0
votes
1answer
147 views

What is the graph of $r \cos \theta = 3$?

What is the graph of $r \cos \theta = 3$? I don't get why there is a $\cos \theta$ in the side of $r$, even if I divide both sides by $\cos \theta$, the right side will be $3/\cos \theta$, which ...
0
votes
1answer
48 views

What is the graph of the polar equation $r = e$?

Is it the same as the graph of $y = e$? A straight line?
0
votes
1answer
511 views

How to calculate the polar arc length of the entire cardioid $r=a(1-\cos\theta)$

I'm having a bit of an issue calculating the arc length of $r = a(1-\cos\theta)$. I'll begin by listing the steps I made in my attempt to solve this exercise. We know that the arc length formula ...
0
votes
1answer
380 views

What is the graph of the polar equation theta = pi?

The question exactly goes like the title. I'm thinking that it's a point on the 3.14, but as I'm typing this I realize that I'm wrong and now I'm out of clues (Google didn't help). Please enlighten ...
1
vote
2answers
125 views

Converting between polar and Cartesian coordinates

The polar coordinates $r$ and $\varphi$ can be converted to the Cartesian coordinates x and y by using the [[trigonometric function]]s sine and cosine: $$x = r \cos \varphi \,$$ $$y = r \sin \varphi ...
0
votes
1answer
274 views

Finding the area bounded by $r = a(1-\sin\theta)$ and $r = a$

Consider the cardioid $r = a(1-\sin\theta)$ and the circle $r = a$. We have that the cardioid meets the origin at an angle of $\frac{\pi}{2}$, while it reaches its maximum distance from the origin at ...
1
vote
1answer
50 views

For which $\alpha \in \mathbb{R}$ does $\int_{\mathbb{R}^n} \big(1+|x|\big)^{\!-\alpha} \mathrm{d}x$ exist?

I assume only $\alpha \gt 1$ gives $\int_{\mathbb{R}^n} (1+|x|)^{-\alpha} \mathrm{d}x \lt \infty$ (simply because this is true for $n=1$). I also assume some clever transformation could be used for ...
0
votes
0answers
15 views

The Set of Closed Curves Representable by $r(\theta)$.

I apologize in advance if my terminology and/or notation is inaccurate; I am a little out of my depth here. If something is unclear, please point it out and I will try to explain myself better. ...
1
vote
2answers
130 views

how to find existence and value of limit in multivariable calculus

I was in maths class and i found a question interesting. Find the limit of $\lim_{(x,y)\to (0,0)} \frac{2x}{x^2+x+y^2}$ if it exist.one of my friend did this question by transforming into polar ...