All Questions
Tagged with integration multivariable-calculus
5,463
questions
2
votes
2
answers
905
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A double integral (differentiation under the integral sign)
While working on a physics problem, I got the following double integral that depends on the parameter $a$:
$$I(a)=\int_{0}^{L}\int_{0}^{L}\sqrt{a}e^{-a(x-y+b)^2}dxdy$$
where $L$ and $b$ are constants.
...
- 6,908
0
votes
2
answers
861
views
Double Integrals
$(a)$ Sketch the region of integration in the integral
$$\int_{y=-2}^{2} \int_{x=0}^{\sqrt{4-y^2}} x e^{{(4-x^{2})}^{3/2}} dx dy$$
By changing the order of integration, or otherwise, evaluate the ...
- 403
2
votes
3
answers
602
views
Evaluating $\int_0^4 \int_{\sqrt {x}}^2 \frac{x^2 e^{y^2}}{y^5}\mathrm dy\,\mathrm dx$
I need help starting with this. I can't find an example like this anywhere in my book
$$\int_0^4 \int_{\sqrt {x}}^2 \frac{x^2 e^{y^2}}{y^5}\mathrm dy\,\mathrm dx$$
- 23
11
votes
2
answers
3k
views
Integral with spherical symmetry over cube
Is it possible to calculate the integral
$$I = \int_{-1}^1 \mathrm dx \int_{-1}^1 \mathrm dy \int_{-1}^1 \mathrm dz \frac{1}{x^2 + y^2 + z^2}$$
analytically? I tried using spherical coordinates
$$I ...
- 345
1
vote
1
answer
474
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Multivariable calculus double integral to polar coordinates
The task is to note down $\iint_D F(x,y)\mathrm dy\mathrm dx$ lane rows in polar coordinates. And region D is defined by $x^2 + y^2 = ax,\, a > 0 $ and $x^2 + y^2 = by,\, b > 0 $ intersection.
...
- 159
4
votes
1
answer
1k
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Trig substitution for a triple integral
This problem involves calculating the triple integral of the following fraction, first with respect to $p$:
$$
\int\limits_0^{2\pi} \int\limits_0^\pi \int\limits_0^{2} \frac{p^2\sin(\phi)}{\sqrt{p^...
- 419
6
votes
1
answer
5k
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vector calculus - Integral over vector
Our physics prof wrote the following equation:
$\int\frac{\vec{r}}{r^3}d\vec{r} = \int\frac{1}{r^2}dr$
This is logical as long as I argue that $\vec{r}$ and $d\vec{r}$ are parallel, which is why the ...
- 1,089
9
votes
0
answers
904
views
Nasty Integral - Closed form solution?
Any suggestions on how to integrate this beast?:
$$\int_0^{\omega_t}\int_{\omega_t}^f\sin^2\left(\frac{\omega_{12}}{2}\right)\sin^2\left(\frac{\omega_{23}}{2}\right)d\omega_{23}d\omega_{12}$$
where:
...
- 2,509
47
votes
4
answers
10k
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Explain $\iint \mathrm dx\,\mathrm dy = \iint r \,\mathrm \,d\alpha\,\mathrm dr$
It is changing the coordinate from one coordinate to another. There is an angle and radius on the right side. What is it? And why?
I got:
$2\,\mathrm dy\,\mathrm dx = r(\cos^2\alpha-\sin^2\alpha)\,\...
- 5,527
2
votes
1
answer
200
views
Problem with Repeated Integrals
I havent had the time to familiarize myself with Latex quite yet, so please excuse my formatting.
I have attempted the following problem four times and got four completely different answers.
$$\...
- 307
3
votes
1
answer
312
views
Integrating ray casting equation
In some ray casting algorithm I want to integrate the following integral:
$\int_{0}^{Z} \frac{c^x}{\|(\vec{o}+x \vec{d})-\vec{l}\|^3} dx$
$Z$ is a constant until which I want to integrate
$\vec{o}$ ...
- 133
3
votes
4
answers
2k
views
How can I find $\int_{\sqrt2/2}^{1}\int_{\sqrt{1-x^2}}^{x}\frac{1}{\sqrt{x^2+y^2}}dydx$?
My question is ; How can I solve the following integral question?
$\displaystyle \int_{\sqrt2/2}^{1}\int_{\sqrt{1-x^2}}^{x}\frac{1}{\sqrt{x^2+y^2}}dydx$
Thanks in advance,
- 393
6
votes
3
answers
26k
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Applying Green's Theorem
So I'm trying to solve this problem stated like this:
Using Green's Theorem, find the area of the elipse defined by (where $a,b \gt 0$):
$$\frac{x^2}{a^2} + \frac{y^2}{b^2} \leqq 1$$
I'm ...
- 287
10
votes
2
answers
1k
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Examples of using Green's theorem to compute one-variable integrals?
We all know that the complex integral calculus can be useful for computing real integrals. I was wondering if there are any similar example where we can use Green's theorem to compute one-variables ...
- 2,259
3
votes
3
answers
303
views
Multiple integral Issue
I'm given the following exercise:
$\iint\limits_D \exp(x^{2}+y^{2})dA$
And I dont even know where to start, any chance someone could give me a hint?
D is a half circle, given by:
$9\le x^{2}+y^{2}...
Emil Madsen