Tagged Questions
2
votes
1answer
53 views
Area of $\left( \frac{x^2}{9}+\frac{y^2}{25} \right)^2 \le x^2 + y^2$
I've used the modified polar coordinates: $x = 3r \cos \theta$, $y =5r \sin \theta$, which got me to
$$r^2 \le 9 \cos^2 \theta + 25 \sin^2 \theta$$
What now?
3
votes
3answers
113 views
Why is the formula for the area of a cardioid $ \int_a^b \frac{1}{2} r^2 d \theta$
I've seen this expression in many places :$\int_a^b \frac{1}{2} r^2 d \theta$ and was wondering if someone can explain where this came from? I've noticed that it's sometimes explained in conjunction ...
0
votes
2answers
53 views
How do you find the maximum value of $r$ in a polar function?
I have $\, r=\cos\alpha +\sin2\alpha,\quad 0\le\alpha\le\frac{\pi}{2}.$
Do you then find $\dfrac{dr}{d\alpha}$ and let that $=0$ ?
I am after just a few set of instructions.
1
vote
1answer
81 views
Find the area of the shaded region between $r=e^{\theta/2}$ and $r=θ$ .
That's the picture of the shaded region I have to find the area of. I'm totally stuck on this problem mainly because these two curves don't intersect so I'm not sure how to find the bounds of ...
0
votes
1answer
25 views
Determining the correct upper bound for an integral in polar coordinates
This seems super easy. But i am just a little bit stuck here. Haven't done much calculus recently. Can someone help me out real quick?
Thank you in advance!
1
vote
1answer
61 views
Finding area between two polar curves using double integrals
I have a homework question that is asking me to find the area that lies:
Inside the curve $r=2+cos(2\theta)$
But outside the curve $r=2+sin(\theta)$
I think I'm supposed to be using a double ...
4
votes
3answers
83 views
Trying to understand the meaning of symmetry
The picture below is the solution to the following problem as presented in my book:
Find the area of the region that lies inside both curves $$r = 8 + \cos \theta \\r = 8 − \cos θ$$
According to ...
2
votes
1answer
38 views
Line integral of $F = r \times k$ on hemisphere
Exam revision -
Verify Stokes theorem directly by explicit calculation of the surface and line integrals for the hemisphere $r=c$, with $z \geq 0$, where $F = r \times k$ and $k$ is the unit vector ...
0
votes
2answers
55 views
Finding the centroid of a polar curve
The curve is $r = e^{-b\theta}$ where $b > 0$ and $θ \in [0, \infty)$.
I got that the arc length is $\frac{\sqrt{b^2 + 1}}{b}$ (is this correct?), but computing the centroid $(x, y)$ looks awful. ...
3
votes
1answer
227 views
How to calculate the area between 2 polar curves: $r=\frac{4}{2}-\sin\theta$ and $r=3\sin\theta$?
How to calculate the area between 2 polar curves: $r=2-\sin\theta$ and $r=3\sin\theta$?
I know that one curve is a limaçon and the other is a circle. I have them drawn out as well, my only question ...
1
vote
1answer
46 views
Polar coordinate
Let $f(x,y)$ be a differntiable function in $\mathbb{R}^2$ so that
$f_x(x,y)y=f_y(x,y)x$ for all $(x,y)\in\mathbb{R}^2$.
Find $g(r)$ so that $g(\sqrt{x^2+y^2})=f(x,y)$ and $g$ is differentiable in ...
1
vote
1answer
73 views
How to calculate a double integral over a triangle by transforming to polair coordinates & by using a transformation
Let T be the triangel with vetrices $( 0,0 ) , ( 1,0 )\mbox{ and } ( 0,1 ) $. Evaluate the integral :
$$
\iint_D e^{\frac{y-x}{y+x}}
$$
a) by transforming to polar coordinates
b) by using the ...
0
votes
1answer
83 views
Find the extremal on the unit disc
I need help for finding the extremal of:
$$J[u]=\int\int_D (u_x^2+u_y^2) dxdy$$
$D$ is the unit disc i.e. $x^2+y^2 \leq 1.$
My boundary condition is
$$u(\cos\theta, \sin\theta)=\sin(n\theta), \ \ ...
0
votes
2answers
88 views
Help understanding the velocity of polar curves.
I have been studying for the AP BC Calculus exam (see this previous question) and most of the questions that deal with the first derivative in polar coordinates say that if ${dr\over d\theta}<0$ ...
1
vote
2answers
124 views
How know which direction a particle is moving on a polar curve
I have being doing problems from the released AP BC Calcululs Free-Response questions, and I have come to realize that I don't have a very good idea of explain or a deep understanding of how to tell ...
1
vote
1answer
87 views
Finding an argument function in a sinusoidal along a circle
I'm attempting to find a function (in polar coordinates) slightly like the one shown below --- i.e. a function which describes a sinusoidal motion along a circular path.
...
4
votes
3answers
344 views
Writing Polar Equations In Parametric Form
For an example problem, in my textbook, the author wanted to demonstrate how to graph a polar function. Deeming it most convenient, my author took the polar function $r=2\cos 3\theta$, and re-wrote it ...
0
votes
1answer
125 views
How to show the normal density integrates to 1?
How could you show that the normal density integrates to 1?
$$ \int_{-\infty}^{\infty} \frac{1}{\sqrt{2\pi \sigma^2}} e^{-(x+\mu)^2 / \sigma^2} dx = 1 $$
1
vote
1answer
506 views
Cardioid given by the polar equation $r = 1 − \cos(\theta)$
Let $C$ be the cardioid given by the polar equation $r = 1 − \cos(\theta)$ , $−\pi \le \theta \le \pi$.
(a) Find the intersection points of the curve with the line $\theta = \pi/4$.
(b) Find the ...
1
vote
0answers
145 views
(Calculus 3) Having trouble finding the polar equation of a hyperbola.
Eccentricity e=sqrt(2), and one vertex is located at (2,0).
I do know that if the vertex is located at (2,0), then the directrix is 2 units from the vertex. I am not sure how to find the location of ...
0
votes
1answer
79 views
Find the area of polar coordinates
The questions is to find the area in the bounded region in polar coordinates
$r = \sqrt{\theta}$ from $3\pi/2$ to $2\pi$
Here is what I did:
I got the integral of $\cfrac{1}{2}\theta d\theta$ from ...
0
votes
1answer
150 views
Calculating the gradient of a tangent of a radial function
I'm trying to determine the gradient of a tangent to a curve defined by a radial function $r = f(\theta)$. It's a programming application and the actual function is gigantic but lets say that $r = ...
4
votes
3answers
345 views
Need hint: Prove that $r = a (\sin t) + b (\cos t)$ is a circle, where $ab \neq 0$
"Show that the polar equation $r = a \sin(t) + b \cos(t)$, where $ab \neq 0$, represents a circle, and find its center and radius."
I don't need the answer - I just need a hint to nudge me on. I've ...
3
votes
2answers
101 views
Help? I cannot do the integration by parts correctly??
Can someone please show me how to integrate
$$\int_0^\infty\frac4{\pi b^2}x^2e^{-x^2/b^2}dx\;?$$
please show steps how to integrate this problem. This is what i have so far.
$$\frac4{\pi ...
1
vote
4answers
164 views
Find the length of a curve
Find the length of the following curve:
$r(t) = e^{-t} \sin(t)+e^{-t}\sin(t) i$ for $0 \leq t\leq 1$.
Any ideas?
1
vote
4answers
1k views
Find the area of the region inside the limaçon
I'm struggling to figure out the answer to this:
Find the area of the region inside the limaçon, $r=3 + \sin(\theta)$
Could someone please help me out?
4
votes
1answer
573 views
Area Bounded by Polar Curves
I am answering sample exams for my Calculus class and my attention was caught by the following item.
Set-up the definite integral or sum of definite integrals equal to the area of the region above ...
1
vote
2answers
648 views
Del operator in 2D polar coordinates
I need to show that the del operator in 2D polar coordinates is $\nabla=e_r\partial_r+\frac{1}{r}e_r+\frac{1}{r}e_{\phi}\partial_{\phi}$. I try the following approach:
...
4
votes
2answers
480 views
Find the area enclosed by the loop $r=2(1-\sin\theta)\sqrt{\cos\theta}$
The diagram shows a sketch of the loop whose polar equation is
$$r=2(1-\sin\theta),\qquad -\frac{\pi}{2}\leq\theta\leq\frac{\pi}{2}$$
a)Show that the area enclosed by the loop is 16/3.
...
0
votes
1answer
338 views
Polar equation of cartesian $y = 1 + 3x$
I have no idea at all what to do on this
I got $$\cos^{-1} \left(\frac{r\sin \theta+1}{3} \right) = \theta$$
Which can be
$$\cos^{-1} \left(\frac{r\sin \left(\cos^{-1} \left(\frac{r\sin ...
0
votes
2answers
132 views
Polar equation of $y = 2$
Maybe I do not understand what is going on here but I cannot get the right answer.
$$y = 2 $$
$$y^2 + x^2 = r^2$$
$$4 + 0^2 = r^2$$
$$ r = 2$$
$$y = r \sin \theta$$
$$1 = \sin \theta$$
$$\theta ...
0
votes
2answers
368 views
Single variable integral to polar coordinates?
I took calculus about 2 semester ago, and I'm trying to brush up on polar coordinates.
I integrated $-x^2+3$ from $x = -\sqrt{3}$ to $\sqrt{3}$ and I got $6.93$
Now I tried to convert it to polar ...
2
votes
1answer
393 views
Find the Cartesian equation corresponding to $r = \frac{5}{3-2\cos(\theta)}$
Find the Cartesian equation corresponding to $r =
\frac{5}{3-2\cos(\theta)}$
I got it into the form:
$(x^2 + y^2)(3-2x)^2 = 25$
and can see that maybe the equation of a circle will appear, but ...
1
vote
1answer
354 views
Replace the Cartesian equation $(x-5)^2 + y^2 = 25$ by an equivalent polar equation.
Replace the Cartesian equation $(x-5)^2 + y^2 = 25$ by an equivalent polar equation.
Let $t= \theta$, $r=5$, $x=r\cos t$, $y=r\sin t$.
I began with $x=5\cos t-5=5(\cos t-1)$ and $y=5\sin t$. Is that ...
1
vote
1answer
96 views
Explain why for $r=1-a \cos^2(3\theta)$ the leaves have the same size only in the case $a=1$ and $a=2$
Please explain why for $r=1-a\cos^2(3\theta)$ the leaves have the same size only in the case $a=1$ and $a=2$.
Does anyone have an answer to this please?
1
vote
1answer
60 views
Am I doing this double integral right?
I want to calculate $\iint_R x \ \mathrm{d}A$, where $R$ is the unit disc centered at $(2, 0)$.
First, I made the following substitution: $$x' = x-2$$ $$\mathrm{d}x' = \mathrm{d}x$$ $$ ...
0
votes
1answer
903 views
Question about the limits of integration using polar coordinates
I haven't been able to find an answer to something I've been thinking about.
If you are taking the integral of a circle in polar coordinates you always use the limits for theta as $0$ to $2\pi$. ...
3
votes
2answers
954 views
how to get $dx\; dy=r\;dr\;d\theta$
In polar coordinate how we can get $dx\;dy=r\;dr\;d\theta$?
with these parameters:
$r=\sqrt{x^2+y^2}$
$x=r\cos\theta$
$y=r\sin\theta$
Tanks.
2
votes
2answers
2k views
Polar to Parametric Equation?
I'm struggling with this problem, I'm still only on part (a). I tried X=rcos(theta) Y=rsin(theta) but I don't think I'm doing it right.
Curve C has polar equation ...
4
votes
2answers
4k views
Why is $dy dx = r dr d \theta$ [duplicate]
Possible Duplicate:
Explain $\iint \mathrm dx\mathrm dy = \iint r \mathrm d\alpha\mathrm dr$
I'm reading the proof of Gaussian integration. When we change to polar coordinates, why do we ...
1
vote
2answers
221 views
Trying to plot these points in a polar coordinate system
I started with:
inside $r_1=5 \sin(θ)$ and outside $r_2=2+\sin(θ)$ and was told to sketch curve in the same polar coordinate system
I first set both equal to $0$ and solved to get $\pi$, $2\pi$, and ...
2
votes
2answers
400 views
Did I sketch this polar curve correctly?
The equation is:
$r^2=-4 \sin(2\theta)$
I first made a reference graph in cartesian coordinates using values $\displaystyle \frac{\pi}{4}$, $\displaystyle \frac{\pi}{2}$, $\displaystyle \frac{3 ...
1
vote
2answers
143 views
Sketching a polar curve
Continued off the question I asked earlier, I also have to sketch the curve.
$r^2=−4\sin(2\theta)$
So I have to set up a table of values I'm assuming. How do I know what values to choose for ...
1
vote
2answers
179 views
How to solve a polar equation when $r$ is $r^2$ instead?
I have $r^2=-4\sinθ$
and I'm asked to set $r=0$, then find θ. If I just set $r^2=0$ then I'll get $\sin(2θ)=0$. That doesn't seem right.
Then I'm asked to set $θ=0$ and then find $r$. If I use the ...
2
votes
1answer
489 views
Polar Coordinates and Double Integrals
Problem 1:
Find the area enclosed by the ellipse $\displaystyle \frac {1} {r} = 1 – 0.6 \cos(\theta)$.
We know $0\leq \theta\leq 2\pi$.
We know $0\leq r\leq 1/(1-0.6\cos(\theta))$.
Questions:
...
24
votes
4answers
2k views
Explain $\iint \mathrm dx\mathrm dy = \iint r \mathrm d\alpha\mathrm dr$
It is changing the coordinate from one coordinate to another. There is an angle and radius on the right side. What is it? And why?
I got:
$2\mathrm dy\mathrm dx = r(\cos^2\alpha-\sin^2\alpha)\mathrm ...
0
votes
1answer
166 views
$\int_{0}^{6} \int_{0}^{y} x dx dy$ where $x = r \cos \theta, y = r \sin \theta, dx dy = r dr d \theta$
Given $x = r \cos \theta, y = r \sin \theta, dx dy = r dr d \theta$, how can I evaluate the following integral:
$\int_{0}^{6} \int_{0}^{y} x dx dy$
6
votes
1answer
569 views
polar coordinates and derivatives
Using the standard notation $(x,y)$ for cartesian coordinates, and $(r, \theta)$ for polar coordinates, it is true that $$ x = r \cos \theta$$ and so we can infer that
$$ \frac{\partial x}{\partial ...
1
vote
2answers
551 views
Difficulty with differentiation in polar coordinates
I am trying to understand a simple calculation in polar coordinates - and I am getting totally discombobulated (the original source can be found: here).Please have a look at the following picture:
...
