0
votes
2answers
42 views

Evaluate an integral using polar $\displaystyle\int_0^2 \int_{-4\sqrt{4-x^2}}^{4\sqrt{4-x^2}}(x^2-y^2)\,dy\,dx$

How do you evaluate the following integral using polar cordinates. $$\int_0^2 \int_{-4\sqrt{4-x^2}}^{4\sqrt{4-x^2}}(x^2-y^2)\:\mathrm{d}y\:\mathrm{d}x$$ I converted it to polar coordinate making it ...
3
votes
1answer
21 views

Finding Multivariable Limits

Is there any good way to find a multivariable limit other than switching to polar coordinates? For example, students each year are inundated with problems like $$\lim_{(x,y)\to ...
1
vote
1answer
29 views

Polar equation — find area under graph using double integral

What is the area of the region in the plane bounded by the curve given in polar coordinates $r = 4 + 2\cos(2\theta)$? Could someone walk me through the conversion of polar coordinates to rectangular ...
0
votes
2answers
24 views

Use Polar Coordinates to Find the Limit…

Use polar coordinates to find the limit. [If $(r, \theta)$ are polar coordinates of the point $(x, y)$ with $r \geq 0$, $r \to 0^+$ as $(x,y) \to (0,0)$)] $$\lim \limits_{(x,y) \to (0,0)} ...
1
vote
1answer
41 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 ...
3
votes
1answer
60 views

Change to polar coordinates when evaluating limits of functions in two variables?

I have a function in two variables $f(x, y)$ and need to calculate the limit $$ \lim_{(x, y) \rightarrow (2, 3)}{f(x, y)} .$$ If I decide to change to polar coordinates, how can I determine where $r$ ...
0
votes
1answer
31 views

Velocity of a particle in polar coordinates

The equations $r = 3\sin(2\theta)$ and $\frac{d\theta}{dt} = 2$ describe the motion of a particle in polar coordinates. Find the velocity of the particle in terms of the unit vectors $u_r$ and ...
1
vote
2answers
61 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 ...
2
votes
1answer
48 views

Area of a sphere bounded by a paraboloid

I need to find the area of the surface $x^2+y^2+z^2 = a^2$ for $y^2 \ge a(a+x)$. I know that $A = 4a \int_{-a}^0 dx \int_{\sqrt{a^2+ax}}^{\sqrt{a^2-x^2}} \frac{dy}{\sqrt{a^2-x^2-y^2}}$, but I have ...
2
votes
1answer
57 views

Help with Polar coordinates and the length of the curve.

I have a test coming up today and I was going over our past midterms and this question came up. I tried it but its not working, please any hints or solution in how to do it will be really helpful. ...
2
votes
1answer
28 views

Polar Integral Confusion

Yet again, a friend of mine asked for help with a polar integral, we both got the same answer, the book again gave a different answer. Question Use a polar integral to find the area inside the ...
4
votes
2answers
59 views

Given integral $\iint_D (e^{x^2 + y^2}) \,dx \,dy$ in the domain $D = \{(x, y) : x^2 + y^2 \le 2, 0 \le y \le x\}.$ Move to polar coordinates.

Given integral $\iint_D (e^{x^2 + y^2}) \,dx \,dy$ in the domain $D = \{(x, y) : x^2 + y^2 \le 2, 0 \le y \le x\}.$ Move to polar coordinates. First of all I tried to find the domain of $x$ and ...
1
vote
2answers
365 views

Volume of a sphere (r=2a) with hole(r=a) drilled through centre, using spherical polar coordinates.

Need help solving 11.bi), A cylindrical hole of radius a is bored through the center of a sphere of radius 2a. Find the volume of the remaining material, using spherical polar coordinates. (You ...
0
votes
3answers
96 views

Changing from Cartesian coordinates to Polar coordinates

Rewrite the iterated integral $$\int_0^1 \int_0^{\sqrt{2y - y^2}} (1 - x^2 - y^2)\,dx\,dy$$ in polar coordinate form. Do not evaluate the integral. Here is my answer: ...
3
votes
7answers
848 views

Why does $r=cos\theta$ produce a circle?

I am trying to do a double integral over the following region in polar coordinates: I know that the limits of integration are: $$\theta=-\pi/2\quad to\quad \theta=\pi/2\\r=0\quad to\quad ...
2
votes
1answer
82 views

Use polar coordinates to find the volume of the given solid

Use polar coordinates to find the volume of the given solid bounded by the paraboloid $z=1+2x^2+2y^2$ and the plane $z=7$ in the first octant. I did it. Is that right ? $$\int_0^{\pi \over 2} ...
1
vote
0answers
34 views

Polar coordinates: fixing ratio between arc and radius

When drawing a coordinate system with fixed step size, the standard polar coordinates $$ x=r\cos(\theta), y=r\sin(\theta) $$ exhibits stretched pixels for large $r$. Ignoring the singularity in ...
1
vote
0answers
126 views

How to plot a stream function

This question relates to fluid mechanics and I have the components in polar coordinates. The components of the velocity field are; $$v_r= \frac{-kr}{z}$$ $$v_z= kz$$ $$v_\theta= 0$$ and I have ...
0
votes
0answers
151 views

Laplace in Polar Coordinates

I'm working on the Laplacian in Polar conversion form Cartesian and I'm having trouble understanding an operation found in the description at ...
2
votes
0answers
78 views

Polar Coordinates for Multivariate Limits With Three Variables

When working on limits with two variables, $f(x,y)$, I like to convert the problem to polar coordinates a lot of the time, by changing the question from $$\displaystyle\lim_{(x,y)\to (0,0)}f(x,y)$$ to ...
0
votes
1answer
47 views

change of variables while integrating

Suppose I have an integral that looks like: $$I=\int_{r=0}^\infty\int_{\omega_1=-\infty}^\infty\int_{\omega_2=-\infty}^\infty ...
1
vote
2answers
37 views

Find the image of a ring

I'm working on the following problem: Find the image of the ring defined by $4 \lt x^2 + y^2 \lt 16 $ under the mapping $$F(x,y) = \left(\frac{x}{x^2+y^2} , \frac{y}{x^2+y^2}\right)$$ It looks to ...
2
votes
1answer
131 views

How do I define the limits of a double integral in polar coordinates over an annulus?

Evaluate the double integral by re-writing them in polar coordinates: $\displaystyle\iint\limits_{R}\frac{y^2}{x^2}\ dA$, where $R$ is part of the annulus (ring) $9\leq x^2+y^2\leq 25$ lying ...
2
votes
1answer
128 views

How to integrate over polar coordinates

Evaluate the following double integral by rewriting it in polar coordinates: $\displaystyle\iint\limits_Dxy\,dA$, where $D$ is the disc with center at the origin and radius 5 I have very little ...
1
vote
0answers
148 views

Polar Integration over intersection of two circles

Let $C_0$ denote a circle centered at $(0,0)$ with a radius of $r_0$ and let $C_1$ denote a circle of radius $r_1$ centered at a point $(x_1,0)$. Assume that we are given some function, $\phi(r)$ ...
2
votes
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?
3
votes
3answers
749 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 ...
1
vote
1answer
436 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 ...
1
vote
2answers
93 views

$f(x,y)=\langle y- \cos y, x \sin y\rangle$

$f(x,y)=\langle y-\cos y,x\sin y\rangle$ $C$ is the circle $(x-3)^2 + (y+4)^2 = 4$ orientated clockwise. Relevant theorems: Green's theorem (this is under the Green's theorem section of our book). ...
1
vote
1answer
160 views

Moment of inertia of a circle

A wire has the shape of the circle $x^2+y^2=a^2$. Determine the moment of inertia about a diameter if the density at $(x,y)$ is $|x|+|y|$ Thank you
2
votes
1answer
101 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 ...
1
vote
1answer
164 views

Polar Coordinates: Dividing by the variable “r.”

Evaluate the iterated integral by converting to polar coordinates: $\large \int^2_0 \int^{\sqrt{2x-x^2}}_0 xy~dy~dx$ I successfully completed most of the problem; however, I had difficulty ...
1
vote
1answer
79 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
0answers
170 views

gradient of an axis symmetric vector field in cylindical coordiantes

I am trying to calculate with a general approach the gradient of an axis symmetric vector field in cylindrical coordinates and then express it in cartesian coordinates. First I write my vector ...
0
votes
1answer
61 views

Polar form $\frac{dy}{dx}$

Trying to find the derivative $\dfrac{dy}{dx}$ in polar form, where: $$x=r\cos\theta \,\text{ and } \, y=r\sin\theta$$ Seems like the common approach (on Wikipedia and other sites) is to assume that ...
2
votes
1answer
139 views

Really Stuck on Partial derivatives question

Ok so im really stuck on a question. It goes: Consider $$u(x,y) = xy \frac {x^2-y^2}{x^2+y^2} $$ for $(x,y)$ $ \neq $ $(0,0)$ and $u(0,0) = 0$. calculate $\frac{\partial u} {\partial x} (x,y)$ and ...
2
votes
0answers
578 views

Parametrization of square to calculate Dot-product in line-integrals and area-integrals, electric field from $\frac{dB}{dt}$?

I am calculating 3A of Tfy-0.1064 in Aalto University. I realized here that I am misunderstanding something in vector calculus: the thing market in green particularly. I know $$\nabla\times E= ...
1
vote
2answers
290 views

Changing Variables in double integral

I have these particular exercise that i cannot solve. I know i have to change the variables, but i cannot figure out if i should use polar coords or any other change. Let D be the region delimited ...
3
votes
1answer
290 views

Stokes' and Divergence Theorem Problems

I have 2 questions on stokes and divergence theorem each. I think I have done both and I just want to make sure that I did them correctly. Question 1 Let $C$ be the boundary of the surface ...
1
vote
0answers
124 views

Explain this limit of integration for radius in polar coordinates.

Use polar coordinates to find the volume of the given solid: Inside both the cylinder $x^2 + y^2 = 4$ and the ellipsoid $4x^2 + 4y^2 + z^2 = 64$ The limit of integration for the radius goes ...
1
vote
0answers
735 views

Mass and center of mass of lamina in polar coordinates

I need some help with the following problem which is question number 15.5.4 in the seventh edition of Stewart Calculus. Here is the problem definition: "Find the mass and center of mass of the ...
1
vote
1answer
177 views

Applying Green's Theorem

So I'm practicing a few problems and I can't get this one - $$P(x,y) = e^x \sin(y) \\ Q(x,y) = e^x \cos(y)$$ $C$ is the right hand loop of the graph of the polar equation $r^2 = 4\cos(\theta)$ ...
1
vote
0answers
107 views

Circle-Circle intersection coordinate system

Consider two points in the 2D Euclidean plane, the origin $0$ and $x$. One can define a co-ordinate system such that for any point $y$ in the plane, $y$ is parametrized by its distance from $0$, call ...
3
votes
4answers
3k views

find all points for intersection between 2 polar equations

I stumped at one of the exercise in my multivariable calculus textbook. I try to search online but I can't seem to search on how answer no 3 and 4 below is derived. I also plot both of polar ...
2
votes
1answer
59 views

gradient in polar coordinate by changing gradient in Cartesian coordinate

I'm tried to do following and I can't see what went wrong. $$\begin{bmatrix} \hat r\\ \hat \theta \end{bmatrix} = \begin{bmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta ...
2
votes
2answers
131 views

Integral variable substitution using Hausdorff measure

Suppose we have positive density $q$ with "good" qualities (continuity, etc..). I need to calculate this integral: $$\int_B q(\textbf{z}) d \textbf{z},\ \textbf{z} \in \mathbb{R}^d,$$ where $B \subset ...
2
votes
1answer
62 views

Proving that inversions are isometries with respect to the hyperbolic metric.

I'd like to prove that the standard inversion $$(r,\theta)\mapsto\left(\frac{1}{r},\theta\right)$$ is an isometry with respect to the hyperbolic metric on the upper half-plane, and it would be nice to ...
2
votes
1answer
2k views

Transforming the Laplace operator from Polar to Cartesian coordinates

I'm trying to find the error in my logic here. Let's say we are given the Laplace operator in polar coordinates: $$ \frac{\partial^2}{\partial r^2} + \frac{1}{r}\frac{\partial}{\partial r} + ...
2
votes
2answers
192 views

How do I find the limit of $\frac{xy\sqrt{|xy|}}{x^2 + xy + y^2}$ as x and y approach zero?

I am trying to find: $$\lim_{(x,y)\to (0,0)}\frac{xy\sqrt{|xy|}}{x^2 + xy + y^2}$$ I suspect that the limit does exist as the combined power of $x$ and $y$ is higher in the numerator than in the ...
1
vote
1answer
238 views

Questions about Hyperbolic Isometries: The Standard Inversion

I have two questions regarding the inversion across the unit circle in the hyperbolic plane. Recall that the hyperbolic plane is a metric space consisting of the open half-plane $$\mathbb{H}^2 = ...