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Questions tagged [iterated-integrals]

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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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37 views

Show Lebesgue Integrable and Compute the Two Iterated Integrals

(I am working on problems having to do with Fubini's Theorem) Given $α ∈ (0,∞)$, show that the function $(x, y) \mapsto e^{−αxy}\cdot sin x$ is Lebesgue integrable on $(0,∞) × (1,∞)$. Compute the two ...
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1answer
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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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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<...
3
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1answer
76 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
56 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
48 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
56 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
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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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Integration by Parts Within Multiple Integral

In the innermost iteration of a triple integral problem, I've applied two instances of integration by parts, and ended up with$$\int_0^{\pi/6}\int_0^{\pi}x\int_0^1ysin(yz)\ dy\ dx\ dz\ = \int_0^{\pi/6}...
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0answers
34 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
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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
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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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How can I solve the following integrals? [duplicate]

I'm trying to solve these integrals. $\int_R max(x,y) \ dxdy$ $\int_R max(x^2,y) \ dxdy$ where $R=[0,1]^2.$ My idea was splitting $R$ into two disjoint sets, $R1=${$(x,y): 0 \leq ...
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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
75 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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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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20 views

Easier Method Showing Integral Over Region is Bounded

Consider the region $$\Delta_n(y):=\left\{{\bf x} = (x_1,\ldots, x_n)\in\mathbb{R}^n:\sum_{i=1}^n x_i\geq y,\;\; x_i\leq y\;\forall i \right\}. $$ Let $f({\bf x})=\prod_{i=1}^n x_i^{-1-\alpha}$ ...
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2answers
62 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
81 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
57 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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0answers
36 views

Writing a double integral as an iterated integral in polar coordinates

Consider the region $R$ in the first quadrant that is outside the circle $r = 1$ and inside the four-leaved rose $r = 2sin2\theta$. Write the following double integral as an iterated integral in ...
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0answers
23 views

Reordering Triple Iterated Integrals

Let $Q$ be the tetrahedron with vertices (0,0,0), (1,1,0), (0,1,0), and (0,1,1) and consider the triple integral $$\iiint F(x,y,z)\, dV$$ In (C.1) and (C.2) rewrite the integral as an iterated ...
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1answer
56 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
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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
43 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
42 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
52 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
51 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
17 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
65 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
42 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 ...
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2answers
70 views

Evaluation of $\int_0^2\int_{y/ 2}^1 ye^{-x^3} dxdy$

Evaluate $$\int_0^2\int_{y/2}^1 ye^{-x^3} dxdy$$ The integral cannot be evaluated by elementary methods, since $e^{-x^3}$ has no elementary antiderivative. However it can be interpreted as $\int\...
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2answers
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Is there any example for $f: I\to \mathbb{R}^n$ both the iterated integrals in Fubini's theorem exists and are equal, yet $f \not \in R(I)$

Reference:(Fubini's Theorem) Question: Is there any example for $f: I\to \mathbb{R}^n$ both the iterated integrals in Fubini's theorem exists and are equal, yet $f \not \in R(I)$ ? Edit: Both ...
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1answer
51 views

Iterated logarithm rule general version exists?

Given a standard random walk, the Iterated logarithm rule say that with probability one, $$\frac{|w(n)|}{\sqrt{n \log\log n}}$$ has $\limsup$ $\sqrt{2}$ as $n \to\infty$. What about other values? ...
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0answers
164 views

Iterated integral exists but $f$ is not integrable - clarification

Consider the function $f$ defined by $$ f(x,y)= \begin{cases} 1 \iff (x,y) =\left(\frac p {2^n},\frac q {2^n}\right): (p,q,n) \in \Bbb N^3, 0<p,q<2^n \\0 \iff (x,y) \neq \left(\frac p {2^n},\...
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0answers
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Graphing the representation of an iterated integral

We have to consider the integral of $$\int_{0}^{1}\int_{0}^{1-x^2} \int_{0}^{1-x}f(x,y,z)dzdydx$$ First we're told to graph the region of integration, so I started that but I'm not exactly sure what ...
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1answer
23 views

Computing an interated R-integral

I want to compute $\int_B f\,\mathrm{d}(x,y)$ with $B=\{(x,y)\in\mathbb{R}^2: |x|<y<a, \ a>0\}$ and $f=ye^{x/y}$. Since we have $|x|<y<a$ I thought to split the integral like this: $\...
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1answer
31 views

Discrepancy between iterated integrals

Question Consider the function $$f(x,y) = \frac{xy(x^2-y^2)}{(x^2+y^2)^3}$$ Using the substitution $u=x^2+y^2$ and $du=2y dy$, the following integral becomes $$\int_0^2 \int_0^1 f(x,y) dy dx = \...
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1answer
261 views

A question on surface integral on how to take projections

While evaluating $$\int\int_S (x^2+y^2)z \ dS$$ where $S$ is the surface of hemisphere $x^2+y^2+z^2=4, z\ge 0$, they have used the formula $$\int\int_S \phi(x,y,z) \ dS = \int\int_D\phi(x,y,f(x,y)) \...
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1answer
49 views

Volume of z = 1-xy using iterated integrals

ok so the question asks for the volume of the function $z = 1-xy$ on the first octant bounded by $x = y$ and $x = y^2 $ i know that on the interval $x = y$ is greater than $x = y^2$ so the limits ...
2
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2answers
75 views

Unsolvable iterated integral: $\int_0^8\int_{y^{1/3}}^2 4e^{x^4} dx\,dy$

Having issues solving the following iterated integral, which appears to not have a tangible antiderivative. $$\int_0^8\int_{y^{1/3}}^2 4e^{x^4} dx\,dy$$ So, I figure it as a graph with the domain of $...
2
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2answers
92 views

Changing order of iterated integral

I am given an integral $$\int_0^1dz \int_0^{1-z}dy\int_0^1f(x,y,z)dx$$ and I want to re-iterate it to have integration with respect to $z$ on the inside and integration with respect to $x$ on the ...
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1answer
40 views

Iterated function composed with initial value of x

I came up with an interesting question for myself but, I got stumbled when trying to solve it. I'm not advanced in maths so forgive me. You are investing in something which gives you back a certain ...
1
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2answers
455 views

Finding area via double integrals, transformation and change of variables.

Find the area of the closed region D formed by the lines x + y = c , x + y = d, y = ax, and y = bx where (0 < c < d, 0 < a < b) by first transforming it into a rectangle with area D' with ...
2
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1answer
86 views

What is meant by $d(x,y)$ in Fubini’s theorem?

According to Wolfram MathWorld, Fubini’s theorem takes a multiple integral over a region $R=\{(x,y):x\in[a,b]\wedge y\in[c,d]\}$ and turns it into an iterated integral by the relationship $$\iint_Rf(...
0
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1answer
56 views

Problems with double integration

I'm trying to integrate $\int_{-a}^a\int_b^cy^{2m+1}e^{xy^{2n}}dxdy$.But I have never seen an integral with so many parts to it and I am little overwhelmed. How do I solve this?
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2answers
81 views

What is the simplest technique to evaluate the following definite triple integral?

Consider the following definite triple integral: $$ \int_0^\pi \int_0^\pi \int_0^\pi \frac{x\sin x \cos^4y \sin^3z}{1 + \cos^2x} ~dx~dy~dz $$ According to Wolfram Alpha, this evaluates to $\frac{\pi^...
3
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1answer
42 views

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\...