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

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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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