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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.

Iterated integral is the process of repeatedly integrating the results of previous integrations.

Integrating one integral is denoted as follows.

Let $~a , ~ b , ~ c ~$ and $~d~$ be numbers and let $~g_1(x) , ~ g_2(x) , ~h_1(y)~$ and $~h_2(y)~$ be functions of $~x~$ and $~y~$ , respectively. Then: $$\int_a^b \int_{h_1(y)}^{h_2(y)} f(x,y)~dy~dx =\int_a^b \left(\int_{h_1(y)}^{h_2(y)} f(x,y)~dy\right)~dx $$and$$\int_c^d \int_{g_1(y)}^{g_2(y)} f(x,y)~dx~dy =\int_c^d \left(\int_{g_1(y)}^{g_2(y)} f(x,y)~dx\right)~dy $$

Note: The order in which the integrals are computed is important in iterated integrals, particularly when the integrand is not continuous on the domain of integration.

References:

https://en.wikipedia.org/wiki/Iterated_integral

https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Calculus_(Apex)/13%3A_Multiple_Integration/13.1%3A_Iterated_Integrals_and_Area

http://math.etsu.edu/multicalc/prealpha/Chap4/Chap4-1/printversion.pdf

141 questions
8
votes
3 answers

Why are these two inequalities not the same even though they use the same equation?

I just don't get it, like at all. $U_{n}$ is an iteration defined on $\mathbb{N}$, BTW. The question was: $$\begin{align} U_{n+1} &= \frac{8U_n - 8}{U_{n} + 2} = 8 - \frac{24}{U_{n} + 2}\\ U_0 &= 3 \end{align} $$ "Prove that $3 \leqslant U_n…
TechnoKnight
  • 435
7
votes
1 answer

Ramanujan: If $\psi(p,n)=\int_0^a\phi(p,x)\cos(nx)dx$, then $\frac\pi2\int_0^a\phi(p,x)\phi(q,nx)dx=\int_0^\infty\psi(q,x)\psi(p,nx)dx$.

Apparently, the following appeared in Ramanujan's first letter to GH Hardy. If $$\psi(p,n)=\int_0^a\phi(p,x)\cos(nx)dx,$$ then $$\frac{\pi}{2}\int_0^a\phi(p,x)\phi(q,nx)dx=\int_0^\infty \psi(q,x)\psi(p,nx)dx$$ I tried proving this, but with not…
clathratus
  • 18,002
6
votes
0 answers

Evaluating $\int_{0}^{1-x} \frac{1}{1-x_{n}} \cdots \int_{0}^{1-x_3} \frac{1}{1-x_2} \int_{0}^{1-x_2} \frac{1}{1-x_1} dx_1 dx_2 \cdots dx_n$

I am trying to evaluate the following integral $$\int_{0}^{1-x} \frac{1}{1-x_{n}} \cdots \int_{0}^{1-x_3} \frac{1}{1-x_2} \int_{0}^{1-x_2} \frac{1}{1-x_1} dx_1 dx_2 \cdots dx_{n}, \hspace{0.5cm} 0
Luciano
  • 93
5
votes
1 answer

Compute $\int_{0}^{1}\int_{0}^{z}\int_{0}^{\sqrt{z^2-x^2}} \dfrac{e^{z^2}}{\sqrt{x^2+y^2}}\, dy\, dx\, dz$ using this change of coorninates.

Compute $\int_{0}^{1}\int_{0}^{z}\int_{0}^{\sqrt{z^2-x^2}} \dfrac{e^{z^2}}{\sqrt{x^2+y^2}}\, dy\, dx\, dz$ using this change of coorninates. I could say that: $0\leq 1, 0\leq x\leq z, 0\leq y \leq \sqrt{z^2-x^2}$ I believe that cylindrical and not…
MonseS
  • 55
  • 3
4
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0 answers

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 variables. I think I was able to solve both of these…
Nick Schemov
  • 123
4
votes
2 answers

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 , but my method was too ugly , could you please offer me…
Tianjian Yang
  • 43
4
votes
2 answers

Find $\int_{0}^{1} \int_{x}^{1}y^4e^{xy^2}dy dx$

$$I:=\int_{0}^{1} \int_{x}^{1}y^4e^{xy^2}dy dx$$ Here the region of integration is the triangle with vertices $(0,0),(0,1)$ and $(1,1)$ and given as a type-1 region. We can convert it into a type-2 region which makes the integral…
ZSMJ
  • 1,264
3
votes
1 answer

Change order of integration for triple integral not possible?

Find the triple integral over the region $E$, where $E$ is bounded by $x=3z^2$ and the planes $x=y, y=0$,and $x=12$. I know that it is possible to integrate in the order $dy$, $dx$, and $dz$ and in the order $dy$, $dz$, and $dx$, but I am having no…
TreeGuy
  • 339
3
votes
1 answer

Double integral of $\frac{x}{1+x^2+y^2}$

I'm trying to crack an integral problem whose answer has been lost: $$ I:=\int_0^\frac{1}{\sqrt{2}}\int_\sqrt{1-x^2}^\sqrt{3-x^2}\frac{x}{1+x^2+y^2}dydx+\int_\frac{1}{\sqrt{2}}^\sqrt{\frac{3}{2}}\int_x^\sqrt{3-x^2}\frac{x}{1+x^2+y^2}dydx$$ Judging…
Boar
  • 357
3
votes
2 answers

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 will also admit also the case in which it is…
Arya
  • 63
3
votes
2 answers

Volumen of the intersection of a cone and a tangent cone.

the professor of our integration class gave us the following exercise as an assignment: Determine the measure in $\mathbb{R}^3$ of $A=\{(x,y,z) \in \mathbb{R}^3 ,x^2+y^2+z^2-2x+2z\leq0\leq x^2-y^2-z^2 \}$ That is, the volumen resulting of the solid…
Javier Herrero
  • 725
3
votes
1 answer

Infinite sum of iterated integrals of matrix products

Edit: Discussion moved to Mathoverflow at https://mathoverflow.net/questions/395085/infinite-sum-of-iterated-integrals-of-matrix-products The problem: Let $$N(z) = \begin{pmatrix} 0 & \frac{1}{z} \\ \frac{1}{2(z-1)^2} & 0 \end{pmatrix} \;.$$ How do…
genus_3_amoeba
  • 31
3
votes
1 answer

Solving a Fredholm Equation of the second kind

I'm trying to solve the Fredholm equation, $$ \phi(x) = 3 + \lambda \int_{0}^{\pi} \text{cos}(x-s) \, \phi(s) \,ds. $$ I began by using the method of successive approximations and found the iterated kernels, $$ \begin{split} K_1 (x,s) &=…
LightningStrike
  • 187
  • 7
3
votes
1 answer

Aren't we summing extra area in this question

This question is rather a simple change of parameters in double integrals, that I am having problem understanding why is it correct. Let $D=\left\{{(x,y)}, ~1\le x^2+y^2\le 4 ~ , x^2+y^2\le2x\right\}$, Find the area of $D$. Now, the question is…
Rab
  • 1,196
3
votes
2 answers

Volume from iterated integrals and two regions

The prompt is to find the volume of the solid which is described the equations and is bounded. $$x^2+y^2+z^2=9 $$ $$x^2-3x+y^2=0 $$ The first one is a sphere with radius 3, the shadow is on the y-x plane. For the second on I tried using completing…
Prathik Gurudatt
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