Questions tagged [iterated-integrals]
This tag is for questions relating to iterated integrals. In calculus, an iterated integral is the result of applying integrals to a function of more than one variable (for example, $~f(x,y)~$ or $~f(x,y,z)~$) in a way that each of the integrals considers some of the variables as given constants.
137
questions
2
votes
1
answer
137
views
$\int_1\dots\int_n\tan^{-1}(x)dx_{1\dots}dx_n$ (Integrating $\arctan$ an Arbitrary Amount of Times)
$$\int_1\dots\int_n\tan^{-1}(x)dx_{1\dots}dx_n$$
I have been trying to solve the above integral, for no reason other than my enjoyment. I am not sure if a closed-form solution exist, or if I am even ...
- 741
0
votes
1
answer
61
views
Change the double integral to an iterated integral
Given:
$$\iint_S (x+y)dA$$
such that the bounded region is given by a triangular area with vertices: $(0,0),(0,4),(1,4)$.
Now we have that $x = \frac{4}{y}$ and $x =0$ for the first iterated integral ...
- 541
2
votes
0
answers
61
views
Evaluate $\int_{0}^{c} \int_{x}^{c} e^{x^2+y^2}dydx$. Given $\int_{0}^{c}e^{s^2}ds = 3$
Evaluate $$\int_{0}^{c} \int_{x}^{c} e^{x^2+y^2}dydx$$
Given $$\int_{0}^{c}e^{s^2}ds = 3$$
where c is a positive constant.
The solution to this question mentioned that:
$$\int_{0}^{c} \int_{x}^{c} e^{...
- 135
0
votes
0
answers
53
views
Perturbative method and Iterated integral
Consider the matrix ode $$\frac{\mathrm d x}{\mathrm d t}(t)=x(t)A(t),x(0)=c$$where $x(t)\in\mathbb R^n, A(t) \in M_{n\times n}$ is continuous. It seems that we can use perturbative method (which I'm ...
- 738
0
votes
1
answer
41
views
Iterated Integral of $(8x^3-36x^2y^2)dydx$
I've been struggling for a bit on this math problem from my homework.
$$
\int_{0}^{1}\int_{1}^{3}(8x^3-36x^2y^2)dydx
$$
From what I understand and have attempted, we'll integrate with respect to $x$ ...
- 23
0
votes
1
answer
32
views
Volume of Same Region Yields Different Values
This question comes from a previous question on this site. There is the integral $$\int_{-1}^{1}\int_{0}^{\sqrt{1 - x^2}}\int_{0}^{\frac{y}{2}}f(x, y, z) \, dz \, dy \, dx.$$ Our job is to reorder the ...
- 270
2
votes
1
answer
70
views
Limits of triple integral over a tetrahedron.
The tetrahedron has vertices $O(0,0,0); A(0,0,2); B(0,2,0); C(1,0,0)$
I was thinking that the plane $ABC$ has equation $2x+y+z=2$ since:
$\vec{BA}= \langle 0,-2,2 \rangle$ and $\vec{CA}= \langle -1,0,...
3
votes
1
answer
143
views
Double integral of $\frac{x}{1+x^2+y^2}$
I'm trying to crack an integral problem whose answer has been lost:
$$ I:=\int_0^\frac{1}{\sqrt{2}}\int_\sqrt{1-x^2}^\sqrt{3-x^2}\frac{x}{1+x^2+y^2}dydx+\int_\frac{1}{\sqrt{2}}^\sqrt{\frac{3}{2}}\...
- 165
3
votes
2
answers
112
views
If all projections on the axis are integrable, do the iterated integrals exist?
Suppose $f(x,y) : [0,1] \times [0,1] \to \mathbb{R}_{\ge 0}$ is a function, not necessarily continuous, nor necessarily Riemann integrable on $[0,1] \times[0,1]$. By existence of an integral here we ...
- 63
0
votes
1
answer
59
views
Find the volume of a body bounded by surfaces
Find the volume of a body bounded by surfaces $z = x^2 + y^2$, $z = x + y$
- 123
1
vote
1
answer
45
views
Find the volume of a body formed by a cylinder and a hyperboloid
Find the volume of a body formed by a cylinder $x^2 + y^2 = 9$ and a hyperboloid $x^2 + y^2 + 9 = z^2$
My solution
Let's try to build the body data
When viewed from above, it will look like a circle ...
- 123
4
votes
0
answers
87
views
Iterated integral change the order and go to polar coordinates
I have an iterated integral with these two homework assignments on it :
(1) change the order of integration
(2) go to polar coordinates and set the limits of integration according to new ones ...
- 123
4
votes
2
answers
153
views
triple integral over a specific region
Evaluate :
$$\begin{equation}
\iiint_{\Omega} z^{2} dV , \quad \Omega : x^{2} + y^{2} + z^{2} \leq R^{2}, \quad x^{2} + y^{2} \leq Rx \quad (R > 0).
\end{equation}$$
I have solved the question , ...
0
votes
0
answers
47
views
Does Fubini's theorem apply on this infinite region?
I came across the following example for a triple integral:
Find the volume of the region bounded by hyperbolic cylinders:
$$ xy = 1 \quad , \quad xy = 9$$
$$ xz = 4 \quad , \quad xz = 36 $$
$$ yz = 25 ...
- 43
0
votes
1
answer
85
views
Can I change the order of integration when the upper limits are infinite?
I'm trying to solve this :
$$\frac{d}{dy}\int_{y}^{\infty} \int_{f(y)}^{\infty} (g(y)+h(x_2))f(x_2)dx_2 f(x_1)dx_1$$
In this case, can I change the order of Integration ? I will get this :
$$\frac{d}{...
- 25