# Tagged Questions

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### 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 ...
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### 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. ...
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### 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 ...
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### Converting polar equation into cartesian equation to obtain derivatives

If I have a polar equation such as $r=1-2\cos(\theta)$ How would I convert this into an equation for $x$ and $y$ so that I can get $dx/d\theta$ and $dy/d\theta$ ?
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### Changing a double integral into polar coordinates

Hi I have the double integral $\int^a_0\int^{\sqrt{a^2-x^2}}_0 e^{-(x^2+y^2+a^2)} dydx$ And I am asked to evaluate this by changing to polar coordinates. I know the transformations are, x=rcosØ ...
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### Cannabis Equation

How can an equation for the following curve be derived? $$r=(1+0.9 \cos(8 \theta)) (1+0.1 \cos(24 \theta)) (0.9+0.1 \cos(200 \theta)) (1+\sin(\theta))$$ (From WolframAlpha)
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### Write function from polar to rectangular coordinates.

I need to write this functions in rectangular coordinates: $$f(r,\theta)=r^{2k+5}\cos5\theta$$ $$g(r,\theta)=r^{2k+5}\cos5\theta$$ Of course the radius is very easy to convert to $x$ and $y$. The ...
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### What is the area outside of $r=1$ and inside $r=2 \cos(3 \theta)$?

What is the area inside the polar curve $r = 2 \cos(3 \theta)$ but outside of the circle given by the polar equation $r = 1$? A picture of the polar curve is at ...
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### Find the area of the region R inside the circle $r=2 \cos \theta$ and outside the cardioid $r=2− 2 \cos \theta$ .

I got $\theta\pi/3$ and $5\pi/3$ and then the area I got was $-4\sqrt3-(8\pi)/3$ The area is not right, I used the area equation that takes integral of $1/2(f(\theta)^2-g(\theta)^2)$ from $\pi/3$ ...
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### Find the area inside a polar curve

I feel a bit silly asking this question as it is no doubt relatively simple, but it has been bugging me. Given the polar curve described by $r^2 = cos(2\theta)$, find the area inside the curve. My ...
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### How would you represent $y=(x-h)^2+k$ in polar coordinates?

I tried using $$x=r\cos(\theta)$$ and $$y=r\sin(\theta)$$ and ended up with $r\sin(\theta) = (r\cos(\theta)-h)^2 + k$ and wasn't sure how to proceed from there.
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### Problems with my work for double integral using polar coordinates

The question is as follows: My work goes like this: ∫∫R sin(x^2 + y^2) dA = ∫(θ from [0, 2π]) ∫(r from [1, 6]) sin(r^2) (r dr dθ) = [∫(θ from [0, 2π]) dθ] * [∫(r from [1, 6]) r sin(r^2) dr] = ...
I'm just learning about polar coordinates now, and I understand the basics pretty well, but I get confused at a particular part. I understand the following relations: $x = r\cos(\theta)$ $y = ... 2answers 23 views ### Polar Equation Conversion Change the polar equation$\theta=\frac{\pi}{3}$to rectangular coordinates. How would I go about this question? I've tried$x=r\cos\theta$and$y=r\sin\theta$, but I can't figure out$r$since ... 1answer 45 views ### Changing operator to polar coordinates Let $$\Delta=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}$$ be the Laplace operator on the$(x,y)$-plane. Consider the polar coordinates with$x=r\cos\theta$and$y=r\sin\theta$. ... 1answer 318 views ### Can someone check my answer for this area between 2 polar curves question? Find the area of the region that lies inside the circle$r = 1$and outside the cardioid$r = 1-cos(\theta)$I drew the graph and set it up like this: $$\int_0^\pi \frac{1}{2} [ (1)^2 - ... 2answers 226 views ### Evaluating the area in the polar coordinates So the problem asked me to find the area of the region that lies inside both of the circles$$r=2sin\theta, \quad r=sin\theta +cos\theta $$I know that r=2sin\theta is x^2+(y-1)^2=1,but ... 0answers 88 views ### Compute \int_{-1}^{1} \int_{\left| x\right| }^{ \sqrt{2-x^2} } \frac{1}{\left( x^2 + y^2\right)^{1000} } \mbox{d}y \mbox{d}x Compute$$ \int_{-1}^{1} \int_{\left| x\right| }^{ \sqrt{2-x^2} } \frac{1}{\left( x^2 + y^2\right)^{1000} } \mbox{d}y \mbox{d}x $$We have (by using polar system):$$ \int_{-1}^{1} ... 2answers 98 views ### Double integrals transforming into Polars This is my first post here. I'm reading about double integrals and can't catch how to get the new limits of integration when converting to polar form. $$\left(\int_{-\infty}^{\infty} ... 3answers 83 views ### Polar to rectangular r = 7 I don't follow this at all. I have r = 7 and the formula states x = r \cos\theta y = r\sin\theta but my book gives x^2 + y^2 = 49 this is impossible. It doens't follow the formula at all. ... 3answers 2k views ### Polar curve r = 2\cos \theta -1$$r = 2\cos \theta -1$$I am suppose to find the polar curve of the inner loop. Here is its graph, courtesy of Wolfram|Alpha, I am having trouble working out this polar function on a cartesian ... 1answer 131 views ### Defining a spiral in polar coordinates I'm trying to find a general form for a spiral that fits the following criteria: the inner radius is N, and for any point q on the spiral, the arc length from the start of the spiral to q is ... 1answer 78 views ### Test for symmetry for polar graphs From a calculus book I'm reading: "Unlike the graphs of an equation in x and y, the graph of an equation r=f(\theta) can be symmetric with respect to the polar axis, the line \theta = \pi/2, ... 1answer 78 views ### Need a hint on what's wrong - polar coordinates I'm asked to solve the following$$ \int^2_0 \int^\sqrt{4-y²}_0 \sqrt{4-x^2-y^2} dxdy $$I thought about using polar coordinates: (1) 0 \le x \le \sqrt{4-y^2} is the upper half of a circumference ... 1answer 104 views ### Converting from polar to Cartesian coordinates. I'm looking at some notes that I was given for my Calculus II class on converting from Cartesian to polar coordinates. Now I understand how to solve for r and \theta but I'm looking at how she ... 1answer 64 views ### Another polar integral bounds question. A plane region R is determined by the inequalities y\ge0, y\ge-x, x^2+y^2\le3\sqrt{x^2+y^2}-3x. Sketch the region and find it's area. I have foregone sketching the area and tried to use ... 1answer 94 views ### area between two polar curves I am trying to find the area between the following two curves given by the following polar equations: r=\sqrt{3}\cos\theta and r=1+\sin\theta. I did the following: First, I found the points of ... 0answers 63 views ### Obtaining the cardioid by mirroring the square root function in a line In what line of the plane C_{W} is the cardioid$$p= 2 (1 + \cos\theta)$$mirrored, from the branch of the function$$w=\sqrt{Z}$$which takes positive values in X>0 and Y=0. Seriously this ... 1answer 60 views ### Area of \left( \frac{x^2}{9}+\frac{y^2}{25} \right)^2 \le x^2 + y^2 I've used the modified polar coordinates: x = 3r \cos \theta, y =5r \sin \theta, which got me to$$r^2 \le 9 \cos^2 \theta + 25 \sin^2 \theta$$What now? 3answers 565 views ### Why is the formula for the area of a cardioid \int_a^b \frac{1}{2} r^2 d \theta I've seen this expression in many places :\int_a^b \frac{1}{2} r^2 d \theta and was wondering if someone can explain where this came from? I've noticed that it's sometimes explained in conjunction ... 2answers 62 views ### How do you find the maximum value of r in a polar function? I have \, r=\cos\alpha +\sin2\alpha,\quad 0\le\alpha\le\frac{\pi}{2}. Do you then find \dfrac{dr}{d\alpha} and let that =0 ? I am after just a few set of instructions. 1answer 395 views ### Find the area of the shaded region between r=e^{\theta/2} and r=θ . That's the picture of the shaded region I have to find the area of. I'm totally stuck on this problem mainly because these two curves don't intersect so I'm not sure how to find the bounds of ... 1answer 32 views ### Determining the correct upper bound for an integral in polar coordinates This seems super easy. But i am just a little bit stuck here. Haven't done much calculus recently. Can someone help me out real quick? Thank you in advance! 1answer 521 views ### Finding area between two polar curves using double integrals I have a homework question that is asking me to find the area that lies: Inside the curve r=2+cos(2\theta) But outside the curve r=2+sin(\theta) I think I'm supposed to be using a double ... 3answers 319 views ### Trying to understand the meaning of symmetry The picture below is the solution to the following problem as presented in my book: Find the area of the region that lies inside both curves$$r = 8 + \cos \theta \\r = 8 − \cos θ$$According to ... 1answer 92 views ### Line integral of F = r \times k on hemisphere Exam revision - Verify Stokes theorem directly by explicit calculation of the surface and line integrals for the hemisphere r=c, with z \geq 0, where F = r \times k and k is the unit vector ... 2answers 279 views ### Finding the centroid of a polar curve The curve is r = e^{-b\theta} where b > 0 and θ \in [0, \infty). I got that the arc length is \frac{\sqrt{b^2 + 1}}{b} (is this correct?), but computing the centroid (x, y) looks awful. ... 1answer 653 views ### How to calculate the area between 2 polar curves: r=\frac{4}{2}-\sin\theta and r=3\sin\theta? How to calculate the area between 2 polar curves: r=2-\sin\theta and r=3\sin\theta? I know that one curve is a limaçon and the other is a circle. I have them drawn out as well, my only question ... 1answer 73 views ### Polar coordinate Let f(x,y) be a differntiable function in \mathbb{R}^2 so that f_x(x,y)y=f_y(x,y)x for all (x,y)\in\mathbb{R}^2. Find g(r) so that g(\sqrt{x^2+y^2})=f(x,y) and g is differentiable in ... 1answer 108 views ### How to calculate a double integral over a triangle by transforming to polair coordinates & by using a transformation Let T be the triangel with vetrices ( 0,0 ) , ( 1,0 )\mbox{ and } ( 0,1 ) . Evaluate the integral :$$ \iint_D e^{\frac{y-x}{y+x}} $$a) by transforming to polar coordinates b) by using the ... 1answer 110 views ### Find the extremal on the unit disc I need help for finding the extremal of:$$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), \ \ ... 2answers 130 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${dr\over d\theta}<0$... 2answers 273 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 ... 1answer 127 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. ... 3answers 1k 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 ...
How could you show that the normal density integrates to 1? $$\int_{-\infty}^{\infty} \frac{1}{\sqrt{2\pi \sigma^2}} e^{-(x+\mu)^2 / \sigma^2} dx = 1$$