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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.

Iterated integral is the process of repeatedly integrating the results of previous integrations.

Integrating one integral is denoted as follows.

Let $~a , ~ b , ~ c ~$ and $~d~$ be numbers and let $~g_1(x) , ~ g_2(x) , ~h_1(y)~$ and $~h_2(y)~$ be functions of $~x~$ and $~y~$ , respectively. Then: $$\int_a^b \int_{h_1(y)}^{h_2(y)} f(x,y)~dy~dx =\int_a^b \left(\int_{h_1(y)}^{h_2(y)} f(x,y)~dy\right)~dx $$and$$\int_c^d \int_{g_1(y)}^{g_2(y)} f(x,y)~dx~dy =\int_c^d \left(\int_{g_1(y)}^{g_2(y)} f(x,y)~dx\right)~dy $$

Note: The order in which the integrals are computed is important in iterated integrals, particularly when the integrand is not continuous on the domain of integration.

References:

https://en.wikipedia.org/wiki/Iterated_integral

https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Calculus_(Apex)/13%3A_Multiple_Integration/13.1%3A_Iterated_Integrals_and_Area

http://math.etsu.edu/multicalc/prealpha/Chap4/Chap4-1/printversion.pdf

137 questions
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Fubini's theorem: Why is the $y$-cut called $y$-cut?

In Fubini's theorem there's the function $f_y$, which is the "$y$-cut" of function $f$. For $I=[a,b] \times [c,d]$ $$f_y:[a,b] \rightarrow \mathbb{R}, f_y(x)=f(x,y)$$ and then $$F:[c,d] \rightarrow \mathbb{R}, F(y)=\int_a^b f_y(x)dx = \int_a^b…
mavavilj
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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 $9$.
Umair Tahir
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How to compute integral, where the bounds given is $x \leq y^2$?

I'm given a function that's piece-wise defined for $x \leq y^2$ and "otherwise". I.e. $$f(x,y) = \begin{cases} x+y^3 & x \leq y^2 \\ 2xy^2 +1 & \text{otherwise} \end{cases}$$ How do I form the integration bounds using this information?
mavavilj
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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 $$\int_0^2 \int_0^1 f(x,y) dy dx = \frac{1}{5}$$ Using the substitution $u=x^2+y^2$ and…
mathnoob123
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