Double integral $\iint f(x,y)$ when $f(x,y)=g(x)+h(y)$.

Question

If I have a function $f(x,y)$ that can be written as $g(x)+h(y)$ can I say that $$\int_a^b \int_a^{b} f(x,y) \,dx\,dy = \int_a^b g(x) \,dx + \int_a^b h(y) \,dy$$

So for Example, $\int_0^1 \int_0^{1} (x + y) \,dx\,dy$ = $\int_0^1 x \,dx$ + $\int_0^1 \,y dy $? When I tried to calculate the values I found put that they don't match. I just want to know why it is wrong.

Answer 1

You might accidentally get away this approach depending on the function and the limits of integration but in general that is not how you would do it. See here for a list of examples.

http://tutorial.math.lamar.edu/Classes/CalcIII/IteratedIntegrals.aspx

Answer 2

\begin{eqnarray} \int_a^{a+1} \int_b^{b+1} f(x + y) \,dx\,dy &=& \int_a^{a+1} \int_b^{b+1} g(x) \,dx\,dy+\int_a^{a+1} \int_b^{b+1} h(y) \,dx\,dy \\ &=& \int_a^{a+1}\,dy\int_b^{b+1} g(x) \,dx +\int_b^{b+1}\,dx\int_a^{a+1} h(y) \,dy \\ &=& \int_b^{b+1} g(x) \,dx +\int_a^{a+1} h(y) \,dy \end{eqnarray}