# Tagged Questions

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### How to evaluate this double integral?

Let $C$ be the subset of the plane given by $$C \colon= \{ \ (x,y) \in \mathbb{R}^2 \ | -1 \leq x = y \leq 1 \}.$$ Then how to evaluate the double integral $$\int_C \int (x^2+ y^2) dx dy?$$ My ...
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### How do I find a point on the surface of a sphere

How do I find a point on a sphere knowing its radius and center point ? I have a sphere: $$x^2+(y-1)^2+(z+3)^2=16$$ Obviously its center point is $(0,1,-3)$ and its radius is $4$. I am asked to find ...
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### Evaluation of the integral of $e^{-(x^2+y^2)}$ over a disk

Show that $$\renewcommand{\intd}{\,\mathrm{d}} \iint_{D(R)} e^{-(x^2+y^2)} \intd x \intd y = \pi \left(1 - e^{-R^2}\right)$$ where $D(R)$ is the disc of radius $R$ with center $(0,0).$ I ...
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### Calculating the area of a region using a mapping

The region: $\{{(x,y) \mid x^{2} < y < 2x^{2}, 2y^{2}<x<3y^{2}, x > 0, y > 0}\}$ The mapping: $u = y/x^{2}$, $v = x/y^{2}$ I calculated the jacobian to be $\frac 34$ which means ...
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### Use a double integral in polar coordinates to find the area

So the area is just an intersection of two circles Converting the two circles to polar coordinates, I get: $r(r-2\sin\theta)=0$, and $r(r-2\cos\theta)=0$ Ummm so $r =0$ and r = $2\sin\theta$ ...
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### Area of spherical cap with integrals

Given a sphere $S$ of fixed diameter $D$ (or radius $R=D/2$, it will be convenient to have both, I suppose), and a point $P$ on its surface, let's create a ball $B$ of variable radius $r$ with its ...
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### Find $\int_0^4\int_{0}^{4}xy \sqrt{1+x^2+y^2} \,dy\, dx$

I am having a tough time figuring this one out. Any help will be appreciated. do we have to approximate, or can we actually find it
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### Double integration:$\int_0^a \int_0^b e^{max(b^2x^2,a^2y^2)}dydx$

I would be grateful for a little help if someone could help me solve a problem in my textbook. The question is, evaluate $\int_0^a \int_0^b e^{max(b^2x^2,a^2y^2)}dydx$, where $a,b$ are positive ...
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### Average value of function over sphere

Here is another qual problem. Suppose $f:\mathbb{R}^3 \mapsto \mathbb{R}$ is $C^2$, and define the (scaled) average function $A(r)=\int_{S^2} f(rn) \:d\sigma(n)$, where $\sigma(n)$ is the usual ...
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### Changing order of integration in cylindrical coordinates

I'm having a problem in changing order of integration in triple integration, in cylindrical coordinates. I would be grateful for a little help.The question is: Let D be the region bounded below by ...
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### Question about the formal justification of nondimensionalization

Assume I have the following (very simple) problem: $\frac{\partial f}{\partial \theta} =0$ and I want to make a change of variables to make it nondimensional. So, I can write: $F = \frac{f}{f^*}$ ...
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### Boundary under transformation of a closed curve from $R^2\to R^3$

Consider some mapping $\phi: R_{uv} \to S\subset \mathbb{R}^3$ where $R_{uv}\subset \mathbb{R}^2$ and such that it is a simply connected region. We call the boundary of the surface (which we ...
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### Simplifying a Vector Integral

While reading the book - Cercignani, Theory and Applications of Boltzmann Transport Equation (I am not a math student), I found this integral which I am unable to understand. Note that $\xi_i , \xi_l$ ...
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### Arc Length with Vector-Valued Functions, Part B

Consider the path of a particle in a conservative force field represented by the vector-valued function $$r(t)= \left(4(\sin t−t \cos t), 4( \sin t+t \sin t), \frac{3}{2} t^2 \right).$$ A) Find the ...
If $g:[a,b]\to\mathbf R$ is a change of 1D coordinates, then the formula is: $$\int_{g(a)}^{g(b)}\,f(x)\,dx = \int_a^b\,f(g(t))\frac{dx}{dt}\,dt. \qquad\text{(1)}$$ If $T=\{x=f(u,v); y=g(u,v)\}$ ...