Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Is the following statement:

$\int_{x_1}^{x_2}{\int_{y_1}^{y_2}{g(x)+h(y)dxdy}} = \int_{x_1}^{x_2}{g(x)dx} + \int_{y_1}^{y_2}{h(y)dy} $

False in the general case ? Does it hold if $g(x) =|x|$ and $h(y) = |y| $ ?

Thanks in advance

EDIT : answered

share|cite|improve this question
up vote 0 down vote accepted

In general that's false, but a similar result is true:

$ \begin{align} \int_{x_1}^{x_2}\int_{y_1}^{y_2}g(x)+h(y)\ dxdy &= \int_{x_1}^{x_2}\int_{y_1}^{y_2}g(x)\ dxdy + \int_{x_1}^{x_2}\int_{y_1}^{y_2} h(y)\ dxdy \\ &= (y_2-y_1)\int_{x_1}^{x_2}g(x)\ dx + (x_2-x_1)\int_{y_1}^{y_2}h(y)\ dy \end{align} $

share|cite|improve this answer
thank you, now I see it. – baibo May 3 '13 at 19:21
@JavierBadia Consider giving a proof for both equalities. – ThisIsNotAnId May 3 '13 at 19:42
@ThisIsNotAnId: Both follow from the linearity properties of the integral. I didn't include a proof because I thought they were pretty straightforward. – Javier May 3 '13 at 20:35

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.