Suppose I have that:

$\gamma = \int_{a_1}^{a_2} \int_{b_3}^{g(y)}h(x,y)dxdy + \int_{a_2}^{a_3} \int_{b_3}^{b_4}h(x,y)dxdy + \int_{a_3}^{a_4} \int_{f(y)}^{b_3}h(x,y)dxdy + \int_{a_4}^{a_5} \int_{b_2}^{b_3}h(x,y)dxdy$

But, suppose it is also the case that $g(a_2)=b_4$ and $f(a_4)=b_2$

Can I then rewrite the integral as

$\gamma = \int_{a_1}^{a_5} \int_{b_1}^{b_4}h(x,y)dxdy$ ?

Or is it only correct to say

$\gamma = \int_{a_1}^{a_3} \int_{b_3}^{b_4}h(x,y)dxdy+ \int_{a_3}^{a_5} \int_{b_2}^{b_3}h(x,y)dxdy$ ?

Or neither?

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Neither solution is correct in general. – Did Jun 3 '12 at 18:34
I guess thats a third possibility – Greg Jun 3 '12 at 19:01