The problem asks to reverse the order of integration of the iterated integral $\int_0^1 \int_{-y}^y f(x, y) dx dy$. We need to find the limits of integration when we integrate with respect to $y$ first and then with respect to $x$.

1. Problem Description

The problem asks to reverse the order of integration of the iterated integral $\int_0^1 \int_{-y}^y f(x, y) dx dy$. We need to find the limits of integration when we integrate with respect to $y$ first and then with respect to $x$.

2. Solution Steps

The given iterated integral is

$\int_0^1 \int_{-y}^y f(x, y) dx dy$.

This corresponds to the region $S$ defined by

$-y \le x \le y$ and $0 \le y \le 1$.

The inequalities can be rewritten as

$|x| \le y$ and $0 \le y \le 1$.

We have $-y \le x \le y$ and $0 \le y \le 1$. This implies that $-1 \le x \le 1$.

We want to express the region in terms of $x$ first.

Since $|x| \le y$, we have $y \ge |x|$.

Also, $y \le 1$.

Thus, $|x| \le y \le 1$.

Therefore, the new limits are $-1 \le x \le 1$ and $|x| \le y \le 1$.

So the reversed iterated integral is

$\int_{-1}^1 \int_{|x|}^1 f(x, y) dy dx$.

We can split the integral into two parts since $|x|$ is defined piecewise.

If $-1 \le x \le 0$, then $|x| = -x$.

If $0 \le x \le 1$, then $|x| = x$.

The reversed iterated integral can be written as

$\int_{-1}^0 \int_{-x}^1 f(x, y) dy dx + \int_0^1 \int_x^1 f(x, y) dy dx$.

However, the problem just asks to reverse the order of integration. So the answer should be $\int_{-1}^1 \int_{|x|}^1 f(x, y) dy dx$.

3. Final Answer

$\int_{-1}^1 \int_{|x|}^1 f(x, y) dy dx$