I was reading a book and I couldn't follow the steps:

$$ y(x)=\int_0^x du \int_0^u f(z,y(z))dz $$

We can rewrite the above as:

$$ y(x)=\int_0^x f(z,y(z))dz \int_z^x du $$

Usually I draw a picture for these kind of integrals. But it happens to have 2 variables. So I couldn't do it this time.


1 Answer 1


Here's what the region looks like, where $1$ is really $x$.

enter image description here

Note that we have $D = \{(u,z) | (0 < u < x) \wedge (0 < z < u)\}$ Another way to write this is $D =\{(u,z) | (0 < z < x) \wedge (z < u < x)\} $

Hence we have

$$ \iint_D f(x,y(z))\ du\ dz = \int_0^x\int_0^uf(x,y(z))\ dz\ du=\int_0^x\int_z^x f(x,y(z)) du\ dz $$


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