改变积分次序

The region is the invariant

A worked reversal

The classic rescue

Splitting when the reversal kinks

The bounds $0 \le y \le 1$, $3y \le x \le 3$ describe a triangle. What are the reversed bounds?

After reversing, $\int_0^3 \int_0^{x/3} e^{x^2}\, dy\, dx$ evaluates to which value?

You face $\int_0^2 \int_{y^2}^{4} \cos(x^3)\, dx\, dy$. What is the right first move?

In the reversed description of the triangle with bounds $0 \le x \le 3$, $0 \le y \le x/3$, the inner pass $\int_0^{x/3} 1\, dy$ returns what expression, in the form x/a like x/7?

Reverse $\int_0^4 \int_{\sqrt{x}}^{2} \cos(y^3)\, dy\, dx$ and evaluate, giving the exact value in the form sin(a)/b like sin(7)/3.