All Questions
823
questions
0
votes
1
answer
93
views
Variable transformation in the definite integral
recently I encounter a variable transformation problem in the derivation and I did not figure out how it works.
$$\int_0^1\int_0^1\frac{\partial^2}{\partial\rho^2}\{s\rho^2C(\textbf{r}_2,ss^\prime\rho)...
- 41
0
votes
0
answers
29
views
integration by parts $\int_{\mathbb{R}^N} -\Delta u (u-M)\varphi dx$
I'm trying to understand how it integrates by parts the following integral. This belong to the academic paper Stable solutions of $-\Delta u=f(u)$ in $\mathbb{R}^N$ with DOI 10.4171/JEMS/217. ...
- 89
2
votes
1
answer
67
views
Evaluation of the given line integral
Question: Evaluate $\int_{C}$B.dr along the curve $x^{2}$+$y^{2}$=1,$z$= 1 in the positive direction from (0,1,2) to (1,0,2);given
B= (xz²+y)i+(z-y)j+(xy-z)k
The question itself is easy,but I don't ...
- 33
3
votes
1
answer
128
views
how to evaluate $\int_{0}^{1}\int_{0}^{1} \frac{2-x-y}{\sqrt{xy}(1+xy) \log_{\pi}^{2}{xy}} \, dx \, dy$
How to evaluate: \begin{align*}
&\int_{0}^{1}\int_{0}^{1} \frac{2-x-y}{(\sqrt{xy}+\sqrt{x^{3} y^{3}}) \left[ \log_{\pi}^{2}{x} + 4\log_{\pi}{\sqrt{x}} \log_{\pi}{y} + \log_{\pi}^{2}{y} \right]} \, ...
0
votes
2
answers
70
views
0
votes
0
answers
21
views
Interchanging Bounds in Double Integrals (When Bounds are Functions of y or x)
Let me set the scene, we have a domain of a function of two variables $f(x,y)$.
$$D = \{(x,y): a \leq x \leq b, g_1(y) \leq y \leq g_2(y)\}$$
So on the $y$-axis it's bounded by two functions. So when ...
3
votes
1
answer
323
views
Flux calculation - what did I do wrong?
The exercise asks to calculate the flux of $\mathbf{F}=(4x,4y,z^2)$ through the surface $x^2+y^2=25, 0\le z \le 2$
I calculated using Gauss' theorem and I obtained $500\pi$, which is, as the teacher ...
- 185
0
votes
1
answer
68
views
What have I done wrong in this double integral?
The exercise I was solving asks to calculate $$\iint_{D} ydxdy$$ where $D$ is limited by $x=0, x^2+y^2=1, y^2=2x, y\ge0$.
What I have done is to use the graphs of $x^2+y^2=1$ and $y^2=2x$ in order to ...
- 185
1
vote
0
answers
21
views
How do I set boundaries in a multiple definite integral equation?
I've been trying to solve these types of problems for a while now and still can't figure out how to set boundaries for each integral in a multiple integral equation. One of the questions is as follows:...
- 11
1
vote
0
answers
79
views
Change the double integral from cartresian to polar cordinates
I have to solve this following integral,
$$\int_0^{\rm A} \int_{\sqrt{R_g^2 - x^2}}^{\sqrt{R^2 - x^2}+r_0} \arcsin{\left(\frac{R_{\rm g}}{\sqrt{x^2+y^2}}\right)} {\rm d}y {\rm d}x$$
Here ${\rm A}$, $...
- 141
1
vote
0
answers
56
views
Show equivalence of two very long integrals.
I'm trying to show that the following integral $$\int_0^\infty \int_{-\infty}^s f(x \vee s-b) \sqrt{\frac{2}{\pi}}\frac{1}{t^{\frac{3}{2}}}(2s-b)e^{-\frac{(2s-b)^2}{2t}-\mu(b+\frac{\mu t}{2})}db ds$$ ...
- 41
0
votes
5
answers
89
views
Help with integration over a triangular region.
I'm currently trying to wrap my head around double integrals over a triangular region, when you are given the vertices of the triangle. I need to do the integral
$$
\iint_D 2e^{-y-x} \,\mathrm{d}y\,\...
- 21
1
vote
1
answer
61
views
Integrate a sum of trig function under absolute value
Let $n \in \mathbb{N}$, I'm trying to compute an explicit formula for the following integral:
$$
\operatorname{I}\left(n\right) = \int_{\left[0,2\pi\right]^{\,\,n}}\,
\left\vert\rule{0pt}{4mm}\,{\cos\...
- 888
1
vote
0
answers
35
views
Solid bounded by five planes
Consider the solid limited by the three coordinates planes and the planes $x+y=1$ and $x+z = 1$. How can I set the limits so I can integrate in this region?
Ideas: the origin is include so $0\leq x, y,...
- 284
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}...
