I am trying to solve the double integral:
$$\int_{0}^1\int_{1-x}^{\sqrt{(1-x)}}e^{\left({\frac{y^2}{2}}-{\frac{y^3}{3}}\right)}\ dydx$$
by reversing the order of integration, however, I am unsure how to go about doing it. Is it right to say that initially:
$\sqrt(1-x)$$≤y≤(1-x)$ and $0≤x≤1$. After reversing the order, we get $1-y^2≤x≤1-y$ and $0≤y≤1$, hence the reversed order of integration will be:
$$\int_{0}^1\int_{1-y}^{1-y^2}e^{{y^2/2}-{y^3/3}}\ dxdy?$$
Best Answer
Your new integral is fine; careful though, this:
and this:
should be the other way around:
$${1-x} \le y \le \sqrt{1-x} \quad \mbox{and} \quad 1-y \le x \le 1-y^2$$
You have the order right in the integrals though.