# Tagged Questions

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### 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 ...
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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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### 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 $r$ ...
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### 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 ...
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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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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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: ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $r^2 = 4\cos(\theta)$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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 = ...