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

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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### 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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### 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 ...
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 $r(y,z)=\left<5y+z^2,y,z\right>$, ...