# Tagged Questions

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### Integrate $f(x,y)=\begin{cases}(x-1/2)^{-3} &\text{if}, 0<y<|x-1/2| \\ 0 &\text{else} \end{cases}$

$f(x,y)=\begin{cases}(x-1/2)^{-3} &\text{if}, 0<y<|x-1/2| \\ 0 &\text{else} \end{cases}$ I have to integrate it over $E=[0,1]\times[0,1]$ I can integrate the inner integral ...
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### Imaginary part of a triple intragral

I'm only an introductory calculus student(I just finished my high school AP calculus BC course) so I apologize if these questions seem a bit elementary. I'm using Mathematica to perform a triple ...
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### How to evaluate this triple integral?

How would I go about evaluating this integral? I want to change the order of integration but don't know how. $$\int_0^1\int_1^{\Large e^z}\int_0^{\log y}x\ dx\,dy\,dz$$ I'm having difficulty ...
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### How can ${\iint\limits_{D}{{e^{x^2+y^2}}}}dxdy$ be found?

How can ${\iint\limits_{D}{{e^{x^2+y^2}}}}dxdy$ be found, if $D$ is $x$ O $y$ axis? So far I have done it this far: ...
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### Integral of multivariate normal density function

Is anybody know a suited close-form solution for this integral: $$I=\int_{R^n} x_i \cdot x_j \cdot f_N({\bf x},{\bf \mu},{\bf \Sigma}) d{\bf x}$$ where ${\bf x}=\{x_1,\ldots,x_n\}$ and $f_N$ is the ...
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### Evaluate double integral:

Evaluate: $$\iint_{D} \arcsin(x^2+y^2)\, dx\,dy$$ where $D$ is defined by the following polar equation $\rho=\sqrt{\sin \theta}$ and $0\le\theta\le\pi$
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### Why is it that $\int_a^b \int_c^d f(x)g(y)\,dy\,dx=\int_a^b f(x)\,dx \int_c^d g(y)\,dy$?

The title sums it up. It's simple to prove, but I'm wondering if there is a geometric interpretation?
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### How to integrate $e^{-z}$ over the ball $x^2 + y^2 + z^2 \leq 1$

I can't figure out how to obtain the limits for this triple integral since I can't visualize how $e^{-z}$ varies over a sphere in my head.
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### Double integral for $\int_{0}^{1} \int_{-1}^{0} \frac {xy}{x^2 + y^2 + 1}\ dy\ dx$
I'm trying to evaluate this $$\int_{0}^{1} \int_{-1}^{0} \frac {xy}{x^2 + y^2 + 1}\ dy\ dx$$ tried substition $$u = {(x^2+y^2+1)}^{-1} \ \ du = \ln {(x^2+y^2+1)}$$ but du is not found in the ...
Find the integration limits of $\int_{0}^{3} \int_{0}^{\sqrt{9 - x^2}} \int_{0}^{\sqrt{9 - x^2 - y^2}} \frac{\sqrt{x^2 + y^2 + z^2}}{1 + (x^2 + y^2 + z^2)^2} dz dy dx$ in spherical coordinates. ...