# Tagged Questions

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### Double integral of $\dfrac{y}{x^2y^2+1}dx~dy$

I'm trying to solve the double integral $\displaystyle\int_0^1\int_0^1\dfrac{y}{x^2y^2+1}dx~dy$ . I'm guessing something with natural log will have to be done. Doing the steps of this problem are more ...
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### Multiple integral 3 dimension

Find the volume of the body $$v:{(x,y,z) :\quad x^2+y^2\le z \le \sqrt{2-x^2-y^2}}.$$ I really don't know what to beside that i have to do triple integral of one. My main problem is to ...
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### Multiple integral what am i doing wrong?

$$\iint _d (x-y)^2*{{{e^y}^+}^z} \qquad d:x-y\ge -1,x-y\le1,x+y\ge1,x+y\le3$$ i was trying to separate it to two multiple integral integrals but i cant integrate this integral, i was trying to ...
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### Evaluate the integral $\iiint\limits_E x^2 \,\, \mathrm{d}V$

Where E is the region bounded by the xz-plane and the hemispheres $y=\sqrt{9-x^2-z^2}$ and $y=\sqrt{16-x^2-z^2}$. This is an exercise from my professor guide. What I tried so far: These exercise ...
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### Arc length for a function $f:\mathbb{R}^2 \to \mathbb{R}^2.$

Assume $f:\mathbb{R}^2 \to \mathbb{R}^2$ is $C^1.$ Is there a formula for the length of the subset of $\mathbb{R}^2$ given by $$\{f(x,y) \in \mathbb{R}^2:a_1\leq x\leq b_1, a_2\leq y \leq b_2\} ?$$ ...
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### Changing order of integration (multiple integral)

Prove $$\int_0^a\left( \int_0^x \left( \int_0^y \left( \int_0^z f(u) \, du \right) dz \right) dy \right) dx = \int_0^a \frac {(a-t)^3}{3!} f(t) dt$$ where $a$ is constant. So I began with two ...
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### Find $\iiint_E sin^3 x+\tan y+ 6\hspace{1mm} dV$, where $V$ is region inside $x^2+y^2+z^2 = 1$

I guess that the integral of $\sin^3 x+\tan x$ part is zero, because i have seen many problems like these where the integral is over a symmetrical region and the functions are odd. But I want ...
### Gauss theorem: $\vec{v} = (x^2 + ye^z, y^2 + ze^x , z^2 + x e^y)$
If D is the region bounded by the cylinder $x^2 + y^2 = 1, z=0$ and $z= x+2$, use the Gauss Theorem to evaluate $$\iint_S \vec{v} . \vec{n} \; dS$$ where $S$ is $\partial D$ , $\vec{n}$ points to ...