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.

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Converting an iterated integral into a **sum** of iterated integrals

Question: Suppose $f$ is integrable. Express $\int^1_0 \int^{2y}_y \int^{x+y}_0 f(x,y,z)\ dz\ dx\ dy$ as a sum of iterated integrals in the order $\ dx \ dy \ dz$. My Attempt: Since $z=x+y$, $z_{max}=...
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continuity of a function and iterated integrals

Let $f:[0,1]\times [0,1] \subseteq \mathbb{R}^2 \rightarrow \mathbb{R}$ defined as : \begin{equation} \label{eq:aqui-le-mostramos-como-hacerle-la-llave-grande} f(x,y) = \left\{ \...
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1answer
35 views

I have a question on Lebesgue Iterated Integrals

Let $(X, S, \mu)$ be a finite measure space. Suppose $g: X \rightarrow \mathbb{R}$ is integrable and that $g(x) \geq 0 \ \forall x \in X$. Let $A = \{(x,y) \in X \times [0,\infty) : 0 \leq y < g(x)...
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1answer
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how to use iterated integral to find area of a region? [closed]

Use iterated integral to find area of a region bounded by $y=\sqrt{x}, y=3, x=0$, and $x=4$. sketch the region.
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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}\\ ...
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36 views

Am I putting the right limits of the integral?

Let: \begin{align} r&=\sqrt{a^2 + p^2 - 2ap \cos \theta}\\ s&=a\\ t&=p\\ f(r) &= \text{continuous function of } r\\ g(s) &= \text{continuous function of } s\\ \end{align} ...
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1answer
31 views

Proving the non-integrability of the following function on $(0,1)\times(0,1)$

So, basically, I've been trying for a while to prove that the function $f(x,y)=\frac{x-y}{(x+y)^3}$ is not integrable in $(0,1)\times(0,1)$. That is, I want to prove that the integral of it's absolute ...
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22 views

Problem with the existence of a iterated integral

Let $$ F(x,y)= \begin{cases} 1 &\mbox{ if } x\in\mathbb{Q},\\ 0 &\mbox{ if } x\in\mathbb{R}\setminus\mathbb{Q} \end{cases} $$ Then, $\int_{0}^{1}\int_{0}^{1}F(x,y)dydx$ exists?\ I'm convinced ...
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21 views

Double integrals and complex number

I used a software to integrate the following double integral of a one sheet hyperboloid over a non-rectangular domain : However as you can see I got a complex number whereas I was expecting to get a ...
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1answer
38 views

Showing Iterated Lebesgue Integrals Do Not Coincide

Let $f(x,y) = \frac{x}{1 - y^2}$ on $(-1,1) \times (-1, 1)$. I have two iterated Lebesgue integrals $$I =\int_{-1}^1 \int_{-1}^1 f(x,y)dxdy ~~~~~~~~~~~\text{and}~~~~~~~~~~~ J =\int_{-1}^1 \int_{-...
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1answer
30 views

I am unable to find correct answer after putting limits

$$ \int_0^3\int_0^{\sqrt{9-y^2}}y\,dx\,dy $$ I am able to evaluate the integral correct but when i put the limits on the last integral my calculation returns $0$, but the correct answer should be $9$....
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26 views

A question about iterated integrals

Good morning, I am currently studying some topics involving iterated integrals of $L^\infty([0,1],\mathbb R)$ functions. At some point there's a proof requiring the choice of $v\in L^\infty$ such ...
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Riemann integration of a piecewise function over $E=[0,1]\times [0,1]$

For the following, let $E=[0,1]\times[0,1]$ and let $S$ be the subset of $E$ defined by $$S=\{(x,y)\in E \ | \ x=\dfrac{2j+1}{2^n}, y=\dfrac{2k+1}{2^n}, j,k=0,...,2^{n-1}-1, n\in \mathbb{N}\}$$ Let $...
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0answers
37 views

Why this iterated integral shouldn't start with $\Omega_1$ or end with $\Omega_0$ in order to converge

I have an issue with the following problem involving iterated integrals: Context. Let $(\omega_1, \dots, \omega_n)$ be complex-valued differential 1-forms defined in a real interval $[a,b]$. For all ...
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Given a vector equation, find the surface area

Question: r(u,v) = ucosv i + usinv j + v k for v and u, 0<=v<=7pi, 0<=u<=1 Find the surface area. from Google I got these steps: Find ...
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1answer
36 views

Evaluate the iterated integral

Evaluate the iterated integral: $$\int_0^2 \int_0^1 (2x+y)^8 \,dxdy$$ I tried doing a u-sub by letting u = 2x+y but then once I got to the outer integral I got $$\int_0^2 (2+y)^9/18 - y^9/18 \, dy$$ ...
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1answer
44 views

Converting Iterated Triple Integral from Cartesian to Cylindrical Coordinates

I am trying to convert the following iterated integral from Cartesian to Cylindrical coordinates: $$\int_{{\,0}}^{{\,\sqrt{3}}}{{\int_{{\,y}}^{{\sqrt {6 - {y^2}} }}{{\int_{{\sqrt \frac{{{x^2} + {y^2}}...
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Can we set up the volume of a 3D solid as a sum of two triple integrals?

For example, consider this question: Write a triple integral, including limits of integration, that gives the specified volume: Under the sphere $𝑥^2+𝑦^2+𝑧^2= 9$ and above the region between $𝑦=�...
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1answer
39 views

Is this the correct integral for finding the volume of the region bounded by the planes $𝑧= 3𝑦,𝑧=𝑦,𝑦= 1,𝑥= 1,$ and $ 𝑥= 2$?

The integral I've formed here is: $$\int_1^2 \int_0^1 \int^{3y}_y 1 dzdydx$$ Which evaluates to one. Not sure if I'm thinking of the region correctly, does this integral make sense for this region? ...
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2answers
54 views

Area between two polar curves using iterated integrals?

The question is from a practice exam I am currently trying to do: I am really not sure how to go about this one. In essence, I'd imagine that the idea is to find the area of the greater curve, and ...
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1answer
27 views

Divergence of a double integral of a exponential

Problem: Solve the double integral $$I:=\int _{-\infty}^{\infty} \int _{-\infty}^{\infty} dx dy\ f(x,y),$$ where $f(x,y) = e^{-i a(x - y)}$ if $y > x$ and zero otherwise ($a$ is only a positive ...
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1answer
51 views

Exponential Growth rate of population

The population of an area is $3500$ lacs with the annual growth rate of $2%$ at $t=0$. After this, the population stabilizes at $14000$ lacs. How much time will it take for the population to reach $...
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2answers
42 views

Calculate the iterated integral $\int\int\sqrt{xy(1-x-y)}dxdy$

Calculate the iterated integral $\int\int\sqrt{xy(1-x-y)}dxdy$ where the domain is $D=\left\{(x,y): x\geq0, y\geq0, x+y\leq1\right\}$ I think the range is $0\leq x\leq1$ and $0\leq y\leq{1-x}$. Is ...
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2answers
53 views

Application of the Divergence Theorem with change of variable

Let $S$ be the ellipse $\left(\frac{x}{a}\right)^2+\left(\frac{y}{b}\right)^2 + \left(\frac{z}{c}\right)^2=1,$ with $\vec{n}$ oriented outwards. Compute $\int\!\!\!\int_S \vec{F}\cdot \vec{n}\,dA$ for ...
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1answer
28 views

Wedge of Cheese Iterated Integral

I am stuck on a practice exam problem. A wedge of Manchego can be modeled as the solid region in $R^3$ bounded by the following surfaces: z = 0, z = y, y = 9-$x^2$. Express the volume of this ...
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0answers
96 views

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<...
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1answer
266 views

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 ...
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1answer
60 views

Re-writing $\int_{0}^{1} \int_{0}^{x}\int_{0}^{x+2y} \mathrm dz\,\mathrm dy\,\mathrm dx$ with the $x$ and $y$ directions first

Can you guys tell me, with reasoning, how to write this triple integral in $x$ and $y$ direction first? I found it very hard to do through graphing. $$\int_{0}^{1} \int_{0}^{x}\int_{0}^{x+2y} \...
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1answer
81 views

Estimate for multiple harmonic sum

I am interested in estimating the following family of sums: $$S_k(n) \equiv \sum_{\substack{n_1, \ldots, n_k \geq 1\\n_1 + \ldots + n_k = n}}\frac{1}{n_1\ldots n_k}$$ where $k \geq 1, n \geq 1$. A ...
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1answer
70 views

Differentials and Integration

I have been informed that consecutive differentials in iterated integral problems are actually connected via the exterior product. So the factor $dx\ dy$ in $\int\int x^2\ dx\ dy$ is actually the ...
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2answers
63 views

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 ...
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0answers
56 views

Is there a well defined convolution where Fubini's theorem fails?

Suppose $f$ and $g$ are real functions such that $f*g\in L^1$ Is there an example where Fubini's theorem might fail for the integral of the convolution? That is: $$\int_X(\int_Yf(x-y)g(y)dy)dx\not=\...
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1answer
50 views

Integral $\int_{[1,2]\times[0,\pi]}\log(\sqrt{x})\sin(2y)d(x,y)$

I want to find out if this integral can be calculated (if it exists) $$\int_{[1,2]\times[0,\pi]} \log(\sqrt{x})\sin(2y)~d(x,y)$$ To be honest, I don't know how, but I think that one might has to use ...
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1answer
29 views

Integral $\int_{[0,1]^3}^{} (x^3+y^2)z^{-1}d(x,y,z)$

I want to find out if this integral can be calculated (if it exists) $$\int_{[0,1]^3}^{} (x^3+y^2)z^{-1}d(x,y,z)$$ To be honest, I don't know how, but I think that one might has to use Fubini's ...
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16 views

project three dimensional region on to various two dimensional planes

Here's an example from the Stewart text that I'm struggling with. [For those with the text this is on page 1074 from the 8th edition]. Suppose we are given $\int_0^1\int_0^{x^2}\int_0^y f(x,y,z) ...
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1answer
123 views

Using Fubini's theorem in proving a function is integrable.

I am having a trouble with proving the following: If $f : (0, a) → \mathbb{R}$ is an integrable function and $$g(x) = \int_x^a \frac{f(t)}{t}dt$$ Then show that $g$ is intergrable and $$\int_0^...
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27 views

question about triple integral with spherical coordinates

Find the volume of the solid E which is located outside the circular cone $x^2 + y^2 = (z − 1)^2$ and between the planes $z = 0$ and $z = 2$. This is not hard problem as I know. I am trying to ...
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2answers
70 views

Is $\iint f(x,y)\; dx\;dy$ called an iterated integral or iterated integrals

In my Calculus 4 class we are learning about finding double and triple integrals. For example, one finds a double integral $$ \iint_E f(x,y)\;dA $$ by writing it as $$ \int_a^b\int_c^d f(x,y)\; dx\;...
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1answer
127 views

Help in numerical verification of Chen's identity in rough path theory.

Would anyone please help me to verify Chen's identity as claimed in https://en.wikipedia.org/wiki/Rough_path#Signature for a simple two dimensional path? Consider a path between $a$ to $d$ and the ...
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1answer
88 views

Double Integral Weird Change of order of integration

I have the following double integral $$\int_0^1 \int_{e^x}^{e^{2x}} x \ln(y) dydx$$ I don't know how to plot a graph here so this is my graph attempt based on Wolfram Alpha I guess it should be $$...
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1answer
141 views

Use Green's Theorem to evaluate a line integral

Evaluate the line integral $\int_cy^4\ dx+2xy^3\ dy$ where $C$ is the ellipse $x^2+2y^2=2$. My attempt: First, I need Green's Theorem: $\int_cP\ dx+Q\ dy = \int\int_D\big(\frac{\partial{Q}}{\...
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1answer
43 views

Double Integral as Iterated Integral: Integrand changes and I don't understand reasoning behind it.

I need to compute the integral $\int \int_{R} y e^{xy}dA$ for $[0,3]\times [0,1]$. In the solutions, it has the double integral rewritten as $\int\int_{R} y e^{xy} dA = \int_{0}^{3} \int_{0}^{1} x e^{...
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1answer
45 views

Help for $\int_0^{\pi^2}\int_\sqrt{x}^{\pi} \sin(y^3) \,\mathrm{d}y \,\mathrm{d}x$ [closed]

SOLUTION $$\begin{aligned} I &=\int_\sqrt{x}^{\pi}\int_0^{\pi^2} \sin(y^3) \,\mathrm{d}y \,\mathrm{d}x \\ &=\int_\sqrt{x}^{\pi}\int_0^{\pi^2} \sin(y^3) \,\mathrm{d}x \,\mathrm{d}y \\ &=\...
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2answers
44 views

How do I perform a change in order of integration here?

I have a function $$\int^1_{y=0}\int^1_{x=y}e^{x^2}dx\ dy$$ Which I want to perform a change in order of integration. I have plotted the graph: And it seems it's the area bounded by the y-axis and x-...
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1answer
54 views

Solve: $\iint_{B}\frac{\sqrt[3]{y-x}}{1+y+x}\,dx\,dy$

B corresponds to the triangle with vertices (0,0), (1,0) and (0,1) I believe I am supposed to use variable substitution; however, whenever I can find a way to simplify the function, the integration ...
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1answer
32 views

How can I solve this Iterated integral

domain E I've this integral on the domain E: x[0,1], y=x^2 $\int_E x^3sin(xy)\, dxdy$ I dont know how can I solve this through the two different ways:first integration in dx and then in dy, and ...
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1answer
59 views

Evaluating $\zeta(4)$ using iterated integrals

I'd like to evaluate $\zeta(4)$ using iterated integrals. We already know the numerical answer, so it remains to set up the integral and do some of the steps. From the recipe of Ihara-Kaneko-Zagier ...
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1answer
18 views

Why is this computation of the surface area incorrect?

So I am asked to find the surface area of the part of the curve $$x=5y+z^2$$ that lies between the planes $y=0,y=z, z=0,$ and $z=2$. I parametized my curve as $r(y,z)=\left<5y+z^2,y,z\right>$, ...
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1answer
77 views

Trying to evaluate this triple integral?

So I'm trying to evaluate the triple integral $$\displaystyle \iiint \limits_{R} \displaystyle \frac{1}{((x-a)^2+y^2+z^2)^{1/2}} \mathrm dV$$ for $a>1$ over the solid sphere $0 \leq x^2 + y^2 + z^2 ...
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2answers
43 views

Iterated integral when $f(x,y)$ is unknown

So we have been given the fact that $ f(3,y-1) = (y-1)(y-3) + f(y,y-1)$ and the assumption is $f_x(x,y)$ is differentiable. I then rearanged to make $f(3,y-1)-f(y,y-1)=(y-1)(y-13). $ Then we need to ...