Which of the following are correct and why?

Question

(reposted due to too many typos and errors) I'm now learning about double integrals and I saw in two different places a thing that contradicted each other and I'd like to know which one is correct.

If I have the following integral:

$\int^a_b\int^g_d cf(x)dxdy$

where $c$ is a constant, which one of the following is the correct one?

1: $\int^a_bc\int^g_d f(x)dxdy$

or

2: $c\int^a_b\int^g_d f(x)dxdy$

If the latter is the correct one and I have:

$\int^a_b\int^g_d cf(x)+g(x)dxdy$

which one of the following is the correct one?

3: $c\int^a_b\int^g_d f(x)dxdy+\int^a_b\int^g_dg(x)dxdy$

or

4: $c\int^a_b\int^g_d f(x)dxdy+c\int^a_b\int^g_dg(x)dxdy$

Repost of https://math.stackexchange.com/questions/3715896/which-rules-of-double-integration-are-correct — Matt Samuel, Jun 11 '20 at 19:32
It is considered bad practice to delete and repost, especially since you got an answer. — Matt Samuel, Jun 11 '20 at 19:35
well, due to my errors, the other post wasn't going to get an answer, hence why I reposted — Duarte Arribas, Jun 11 '20 at 19:35
Do not delete and repost questions. If you have typos, edit. — Asaf Karagila, Jun 12 '20 at 09:34

score 0 · Answer 1 · answered Jun 12 '20 at 01:28

I don't see the difference between options (1.) and (2.). For the question of how to distribute $c$ in $\int_a^b \int_d^g [c f(x) + g(x)] \, dx \, dy,$ use the linearity of the double integral to obtain $\int_a^b \int_g^d c f(x) \, dx \, dy + \int_a^b \int_d^g g(x) \, dx \, dy;$ then, use the result in (1.) (or (2.), as they are the same).

P.S. -- I assume that you meant to write $f(x, y)$ and $g(x, y);$ otherwise, by Fubini's Theorem, $$\begin{align*} \int_a^b \int_d^g [cf(x) + g(x)] \, dx \, dy &= \biggl(\int_d^g [cf(x) + g(x)] \, dx \biggr) \biggl(\int_a^b 1 \, dy \biggr) \\ \\ &= (b - a)\biggl(\int_d^g [cf(x) + g(x)] \, dx \biggr). \end{align*}$$

Which of the following are correct and why?

1 Answers1