# Tagged Questions

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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 ...
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### 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 ...
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### Surface Integral of $\frac{\vec{r}}{|\vec{r}|^3}$

Let $$\vec{v} = \frac{\vec{r}}{|\vec{r}|^3},$$ $\vec{r}=(x,y,z)$. Evaluate $$\iint\limits_S \vec{v} \cdot \vec{n} \, dS$$ with $\vec{n}$ pointing to the exterior of the surface $S$ ...
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### Surface integral over a rectangle

Evaluate $$\iint_S \vec{v} . \vec{n} dS$$ for $\vec{v} = (x+y, -2y - 1, z)$ and $S$ the rectangle of vertices $(1,0,1),(1,0,0),(0,1,0),(0,1,1)$ and $\vec{n}$ points in the opposite direction of ...
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### Surface integral over the plane $x+y+z=2$

Evaluate $$\iint_S x\;dy \times dz + y \; dz \times dx + z \; dx \times dy$$ where $S$ is the part of the plan $x+y+z=2$ in the first octant, with normal $n$ such that $n . (0,1,0) \geq 0$ My ...
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### Surface integral over a sphere - parametrization

Evaluate the surface integral of the field $A(x,y,z)=(xy, yz, x^2)$ over the sphere $S$ givn by $x^2 + y^2 + z^2$ with the normal vector pointing to the exterior of the sphere I've tried doing this ...
$$\int_{0}^{2\pi} \int_{0}^{1}r^5\sin^22\theta\left(1-r^2 \right)^2\sqrt{1+\left(1+ \cos^2\theta \right)36r^2 }\hspace{1mm}drd\theta$$ I tried integrating myself, spent many hours but could not ...
$$\int_0^1\int_0^1x^3y^2\sqrt{1+x^2+y^2}\hspace{1mm}dxdy$$ We have to compute this up to 4 decimal places