All Questions
Tagged with integration multivariable-calculus
401
questions
47
votes
4
answers
10k
views
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
9
votes
1
answer
1k
views
The Fourier transform of $1/p^3$
The Fourier transforms we use are
\begin{align}
\tilde{f}(\mathbf{p})&=\int d^3x\,f(\mathbf{x})
e^{-i\mathbf{p}\cdot\mathbf{x}}\\[5pt]
f(\mathbf{x})&=\int \frac{d^3p}{(2\pi)^3}\,\tilde{f}(\...
- 505
16
votes
4
answers
7k
views
Volume of $T_n=\{x_i\ge0:x_1+\cdots+x_n\le1\}$
Let $T_n=\{x_i\ge0:x_1+\cdots+x_n\le1\}$. I know $T_n$ is tetrahedron.
My question: How can I compute the volume of $T_n$ for every $n$?
user145801
21
votes
3
answers
9k
views
General condition that Riemann and Lebesgue integrals are the same
I'd like to know that when Riemann integral and Lebesgue integral are the same in general. I know that a bounded Riemann integrable function on a closed interval is Lebesgue integrable and two ...
- 251
46
votes
5
answers
19k
views
Interpreting Line Integrals with respect to $x$ or $y$
A line integral (with respect to arc length) can be interpreted geometrically as the area under $f(x,y)$ along $C$ as in the picture. You sum up the areas of all the infinitesimally small 'rectangles' ...
- 5,361
7
votes
3
answers
602
views
Find the value of $\int_{-\infty}^\infty \int_{-\infty}^\infty e^{-(x^2+xy+y^2)} \, dx\,dy$
Given that $\int_{-\infty}^\infty e^{-x^2} \, dx=\sqrt{\pi}$. Find the value of $$\int_{-\infty}^\infty\int_{-\infty}^\infty e^{-(x^2+xy+y^2)} \, dx\,dy$$
I don't understand how I find this double ...
- 6,470
71
votes
2
answers
27k
views
Why absolute values of Jacobians in change of variables for multiple integrals but not single integrals?
If $g:[a,b]\to\mathbf R$ is a change of 1D coordinates, then the formula is:
$$ \int_{g(a)}^{g(b)}\,f(x)\,dx = \int_a^b\,f(g(t))\frac{dx}{dt}\,dt.
\qquad\text{(1)}$$
If $T=\{x=f(u,v); y=g(u,v)\}$ ...
- 821
21
votes
4
answers
10k
views
When to differentiate under the integral sign?
I'm finishing up a semester of multivariable calculus and will be taking a course on analysis this Spring. In any of the calculus courses I've taken, we never covered anything beyond the standard ...
3
votes
3
answers
7k
views
How do I calculate a double integral like $\iint_\mathbf{D}e^{\frac{x-y}{x+y}}dx\,dy$?
The Problem is to integrate the following double integral $\displaystyle\iint_\mathbf{D}\exp\left(\frac{x-y}{x+y}\right) dx\, dy$ using the technique of transformation of variables ($u$ and $v$).
...
- 847
3
votes
4
answers
926
views
Calculating volume enclosed using triple integral
Calculate the volume enclosed between $x^2 + y^2=1$, $y^2+z^2=1$, $x^2+z^2=1$
I am supposed to solve this question using multiple integral. I am not able to visualize the resultant figure of whose I ...
user88923
8
votes
3
answers
967
views
How should I solve this integral with changing parameters?
I can't solve this. How should I proceed?
$$\iint_De^{\large\frac{y-x}{y+x}}\mathrm dx\mathrm dy$$
$D$ is the triangle with these coordinates $(0,0), (0,2), (2,0)$ and I've changed the parameters ...
- 139
5
votes
3
answers
877
views
How to solve this integral for a hyperbolic bowl?
$$\iint_{s} z dS $$ where S is the surface given by $$z^2=1+x^2+y^2$$ and $1 \leq(z)\leq\sqrt5$ (hyperbolic bowl)
- 425
0
votes
1
answer
2k
views
Double integral with transformation; possible error in limits
I'm asked to show that the following integral is true with the transformation $u = x+y$ and $y=uv$:
$$\int_0^1 \int_0^{1-x} e^{y/(x+y)} dx dy = \frac{e-1}2 $$
I found the determinant of the jacobian ...
- 115
77
votes
2
answers
9k
views
Integration of forms and integration on a measure space
In Terence Tao's PCM article: DIFFERENTIAL FORMS AND INTEGRATION, it is pointed out that there are three concepts of integration which appear in the subject (single-variable calculus):
the indefinite ...
user9464
44
votes
3
answers
9k
views
How to change variables in a surface integral without parametrizing
This is a doubt that I carry since my PDE classes.
Some background (skippable):
In the multivariable calculus course at my university we made all sorts of standard calculations involving surface ...
- 3,447