# Tagged Questions

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1answer
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### Is there a need for another integration technique?

I'm being asked to calculate $$I\triangleq\int_0^1\int_{e^{\large x}}^e{xe^y\over(\ln y)^2}\,dy\,dx\quad.$$ I got stuck on the indefinite inner one, $$J\triangleq\int{e^ydy\over(\ln y)^2}\quad.$$ At ...
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### The tightest bound on an integral

Consider a polynomial $p(x)$ where $p(x)>0$ for $x\in(0,1)$ and $p(0)=0$. Let $s(x)$ be an increasing analytic function such that $s(0)=0$ and $s(1)=1$. I am interested to bound the following ...
1answer
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### Finding volume under surface and above a region

I'm asked to find $\underset{U}{\int}(x+y)^2\, dA$ where U is a region bounded by the lines x = -1, x = 1, y = -1 ... and by the curves x=$y^2$ , y=1+$x^2$ Plot: http://d.pr/WYSg I started out by ...
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### Finding the centre of mass? What axis does the centre of mass lie on?

Let the mass density $\mu$ be given by $$\mu(x,y,z)= \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} \leq1$$ what axis would the centre of mass lie on?
3answers
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### Confusing Triple Integral

i'm having trouble with this integral the integral is $\int_0^9\int_{\sqrt z}^3\int_0^y z\cos(y^6)\,dx\,dy\,dz$. We aren't given any more information and i'm a bit stuck as to where to start. I don't ...
2answers
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### Triple integral problem involving a sphere

Let $R = \{(x,y,z)\in \textbf{R}^3 :x^2+y^2+z^2\le\pi^2\}$ How do I integrate this triple integral $$\int\int\int_R \cos x\, dxdydz,$$ where $R$ is a sphere of radius $\pi$? I have trouble ...
1answer
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### Integration in $\mathbb{R}^n$ region

If its all parameterized usually I can solve it, but I have a problem with integration in vagues regions, usually I dont know the right procedure to solve them. The problem I need to solve is: given ...
2answers
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### Find integral of a polar function $h(r,\theta)$ over a circle

I am studying for my math final and our prof gave us a review but without any solutions or hints. I don't really understand this problem so if anyone could help me out here I would appreciate it. ...
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### Multivariable weird function

I have to prove two statements, but this function is so weird( and hard to work with )...I just can't figure out how to solve this. The given function is $\varphi:\mathbb{R}^N\rightarrow\mathbb{R}$ ...
1answer
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### 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
1answer
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### Work and Line Integral

A two-dimensional force field is given by the equation $$f(x,y)=cxy\textbf{i}+x^6y^2\textbf{j}$$, where $c$ is a positive constant. This force acts on a particle which must move from $(0,0)$ to the ...
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### Triple Integral Spherical Coordinates

So I have to compute the triple integral of this: $\int\int\int \frac{1}{1+x^2+y^2+z^2}$ and it says the equation of the sphere is $x^2 + y^2 + z^2 = z$ which is just an elongated sphere running ...
1answer
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### tricky surface integral

I am studying for my final and my prof gave us review questions but with no answers so I am lost with this question. If anyone can help I would really appreciate it. Question: Find the area of the ...
0answers
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### Intuitive understanding of integral of vector valued functions

Today in class we were introducing complex line integrals. And that got me thinking, I don't know of a good interpretation for integrals of functions from $\mathbb{R}$ to $\mathbb{R}^2$ or ...
2answers
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### Volume integral over a bounded region

Class is over now and I am studying for my final and I have a problem with this question on our review sheet. If anyone can help I would appreciate it. Question: Find the volume of the region in ...
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### 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 ...
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### Double Integral of piece wise function?

Let $I=[0,1]\times[0,1]$ and let $$f(x)= \begin{cases} 0, & \text{if (x,y)=(0,0)}\\ \frac{x^2-y^2}{(x^2+y^2)^2}, & \text { if (x,y)\not=(0,0)}\\ \end{cases}$$ Need to show that ...
1answer
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### 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 ...
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### volume evaluated by triple integral

Let $\Omega:=\{(x,y,z)|x^2+y^2=1, 0\leq z \leq 2\}$, fix an $\alpha \in (-\frac{\pi}{2},\frac{\pi}{2})$ and given the transoformation $T(x,y,z):=(x,y+z\tan \alpha,z)$, find the volume of $T(\Omega)$. ...
1answer
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### How to integrate a vector function in spherical coordinates?

How to integrate a vector function in spherical coordinates? In my specific case, it's an electric field on the axis of charged ring (see image below), the integral is pretty easy, but I don't ...
2answers
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### Determining the Moment of inertia

Let $a,b,c$ be positive real numbers such that $c<a$. Suppose given is a thin plate $R$ in the plane bounded by $$\frac{x}{a}+\frac{y}{b}=1, \frac{x}{c}+\frac{y}{b}=1, y=0$$ and such that the ...
2answers
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### Double integral of polar coordinates?

Compute $\int_C (8-\sqrt{x^2 +y^2}) ~ds$ where $C$ is the circle $x^2 + y^2 =4$. Answer: $24\pi$ How is the answer $24\pi$? I converted the integral into a double integral of polar coordinates ...
2answers
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### Integral from $0$ to Infinity of $e^{-3x^2}$? [duplicate]

How do you calculate the integral from $0$ to Infinity of $e^{-3x^2}$? I am supposed to use a double integral. Can someone please explain? Thanks in advance.
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### Algorithm to calculate multiple integral.

One of the major difficulties of student in advanced calculus (including myself when student) is to obtain the extremes of repeated integrals to calculate the volume integral in $R^n$ i.e. transform ...
3answers
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