All Questions
137
questions
2
votes
1
answer
66
views
What coordinate substitution should I perform to evaluate this triple integral?
I am trying to evaluate the following triple integral:
\begin{equation}
\int_{-1}^1 \int_{-\sqrt{4-4x^2}}^{\sqrt{4-4x^2}} \int_{\sqrt{4x^2 + z^2}}^2 ye^{4x^2 + y^2 + z^2} \, dy\, dz\, dx
\end{equation}...
4
votes
0
answers
53
views
evaluate the volume of solid
Consider the paraboloid $(\mathcal{P}): z=x^2+y^2$ and the plane $(\mathcal{Q}): 2x+2y+z=2$.
Let $\mathcal{S}$ be the solid region bounded above by $(\mathcal{Q})$ and below by $(\mathcal{P})$. Find ...
- 319
0
votes
2
answers
39
views
set the limits of integration of the spherical coordinates between two paraboloids and a plane
Find the volume of the solid $\mathcal{S}$ enclosed laterally by the paraboloids $\mathcal{P}_1$ of equation $z = x^2 + y^2$ and $\mathcal{P}_2$ of equation $z = 3(x^2 + y^2)$ and from above by the ...
- 319
3
votes
3
answers
90
views
Integrating $e^{-(x^2+y^2+z^2)/a^2}$ over $\mathbb{R}^3$
I want to compute the following integral
$$
\int_{\mathbb{R}^3}e^{-(x^2+y^2+z^2)/a^2}\,dxdydz
$$
and I thought that using spherical coordinates could make it easier, since $r^2=x^2+y^2+z^2$. With this ...
- 3,435
2
votes
1
answer
129
views
Integrating general powers of the Mahalanobis norm with respect to spherical measure
Let $S^{d-1}$ denote the $d$-dimensional sphere, and $\sigma_{d-1}$ denote the corresponding spherical measure. I am wondering how to go about solving the following integral for $c>0$
$$
\int_{S^{d-...
- 6,106
1
vote
1
answer
77
views
Sketching Domain in Spherical Coordinates
$D$ is the portion of the unit cylinder $x
^2 +y
^2 ≤ 1$ which lies between $z = 0$ and $z = 1$.
I normally solve questions like this by sketching the $z-r$ plane. Obviously I draw $z=0$ and $z=1$, ...
- 261
3
votes
0
answers
86
views
Paraboloid and Sphere
Find the volume of the solid region inside the sphere $x^2+y^2+z^2=6$ and above the paraboloid $z=x^2+y^2.$
I set both equations equal to each other and obtained $z=2$ and $z=-3$. Since clearly $z>...
- 261
0
votes
1
answer
171
views
Convert triple integral in cylindrical coordinates to spherical coordinates
I have this math problem right now, and any way I look at it, it just seems impossible. I don't understand.
So I am given the integral:
$$
\int_0^{2\pi}\int_0^{1}\int_0^{\sqrt{4-r^2}} r^2\,
dz\,dr\,d\...
- 103
0
votes
0
answers
46
views
Volume of intersection of two partial spheres in spherical coordinates
Assume I have one partial sphere $s_1$ defined by center $c_1: (x_1, y_1, z_1)$, radius: $\rho_1$, polar angle: $\phi_{1, min} \leq \phi_1 \leq \phi_{1, max}$, and azimuthal angle: $\theta_{1, min} \...
1
vote
1
answer
56
views
Evaluating the triple integral of $ f(x,y,z) = e^{(x^2+y^2+z^2)^\frac{3}{2}} $
as part of a math tutorial I am supposed to evaluate the following integral:
$$ \int_{-1}^1
\int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}}
\int_{-\sqrt{1-x^2-y^2}}^\sqrt{1-x^2-y^2}
e^{(x^2+y^2+z^2)^\frac{3}{2}}
...
- 55
0
votes
1
answer
19
views
How do I evalute the value of an given only the radii of 2 spheres??
The question basically. I was able to find rho and theta $rho$ and $theta$ as $\int_1^5\int_0^2pi$,(2pi is the top limit), respectivley. I can't find $phi$ though and I'm assuming thats wrong. My ...
- 73
1
vote
2
answers
487
views
Variable change in integral with spherical coordinates in $N$-dimensinal space
I am having difficulty with this problem.
Given $f(x)$ is smooth enough, $x \in \mathbb{R}^N$. The spherical coordinate of $x$ is $x = \left(r, \sigma\right)$, $r = \left\|x\right\|$, $\sigma \in S^{N-...
- 89
0
votes
0
answers
47
views
Cartesian coordinates to spherical coordinates
I need to solve this integral $$\int _0^1\:\int _{-\sqrt{1-x^2}}^{\sqrt{1-x^2}}\:\int _{\sqrt{x^2+y^2}}^1\:dzdydx$$ using spherical coordinates (I already made a sketch of the volume).
The problem is ...
- 331
3
votes
1
answer
47
views
Which region is this triple integral reffering to?
$$Integral = \iiint_V x+z\;dV $$
$$Region$$
$$x^2 + y^2 + z^2 \leq 1$$
$$z \leq \sqrt{x^2 + y^2}$$
Geogebra shows that the region looks like in the picture below but given the inquality sign in the ...
- 173
0
votes
1
answer
46
views
What would be the bound of integration for this triple integral?
I don't understand why theta and thy range from 0 to pi/2. shouldn't it be from 0 to pi/4 since you divide the whole sphere by 8? a full circle is 2pi, so eight of a circle should be pi/4. Down below ...
- 785