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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1answer
70 views

$\int_{0}^{1} \int_{0}^{1} \sqrt{x^2+y^2} dxdy$

I would like to solve the given integral: $$\int_{0}^{1} \int_{0}^{1} \sqrt{x^2+y^2} dxdy$$ The integral is doable just using the regular iterated integral with $x$ and $y$. We integrate with respect ...
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50 views

Show the equality between two integrations

$$\int ^{\infty}_{0}\int^{\infty}_{0} e^{-t[1+v]} v^{-s}\,dvdt=\int^{\infty}_{0} \frac{v^{-s}}{1+v}\,dv$$. This shows up in one step of the proof of some property of gamma function in Stein & ...
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0answers
28 views

Greens Theorem & does the perimeter of a double integral over a region converge to the perimeter of the original region?

When I've seen double integrals presented, usually its visualized as adding a bunch of small rectangular dA elements along the region. It feels pretty reasonable that this converges to the area. What ...
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1answer
26 views

Questions using Iterated Integrals.

so Ive just been introduced to the idea of iterated integrals and im finding it hard to work out how to complete questions on this subject and was wondering if anyone could help. So if I have an ...
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1answer
27 views

how to prove that two iterated integral are different

We have the following function: $$f(x,y)= \begin{cases} 1, & 0 \leq x-y \leq 1\\ -1, & 0 \leq y-x \leq 1 \\ 0, & \text{other cases} \end{cases} $$ and I need to prove that $...
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1answer
33 views

How do I get the normal domain from an inequality condition (application of Fubini's Theorem)?

I want to compute the volume of the 3-dimensional unit ball. $A := \{(x,y,z) \in \mathbb{R}^3: x^2 + y^2 + z^2 \leq 1$ From this I want to get the normal domain as: $A = \{(x,y,z): x \in [-1,1], y \in ...
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1answer
70 views

Find the surface area of the sphere inside the cylinder

The given equations are of a sphere and cylinder respectively $$x^2+y^2+z^2=400$$ $$x^2+y^2=256$$ Solving the sphere equation for $z$ yields $$z=\sqrt{400-x^2-y^2}$$ Now to find $ds$ we take the ...
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2answers
133 views

Problem with Infinite number of iterated integrals

I have been working on a problem from a past high school Math Competition and I have been stumped. I am asked to compute $$\lim_{n\rightarrow\infty} \int_{0}^{1}\int_{0}^{1}\cdot\cdot\cdot\int_{0}^{1}\...
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28 views

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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1answer
23 views

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
37 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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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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3answers
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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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38 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
38 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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0answers
28 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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1answer
50 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_{-1}^1 ...
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1answer
35 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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0answers
24 views

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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0answers
52 views

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
41 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
107 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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0answers
25 views

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
59 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
59 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
30 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
78 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
265 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
88 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
31 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
179 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
378 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
61 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
92 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
72 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 ...
3
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2answers
66 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
66 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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0answers
18 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
143 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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0answers
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 ...
1
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2answers
78 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
203 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
103 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 $$\...
2
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1answer
394 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
47 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
46 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 \\ &=\...
2
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2answers
50 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-...