The image shows an equality between two double integrals:

$\int_0^2 \int_{-x}^{x} f(x,y) \, dy \, dx = \int_{-1}^1 \int_{|y|}^{2} f(x,y) \, dx \, dy.$

Explanation:

1. Left Side Integral:

   • The outer integral runs from $x = 0$ to $x = 2$.
   • For each fixed $x$, the inner integral runs from $y = -x$ to $y = x$.
   • This describes a region in the $xy$-plane where $x$ ranges from $0$ to $2$ and $y$ ranges between $-x$ and $x$.

2. Right Side Integral:

   • The outer integral runs from $y = -1$ to $y = 1$.
   • For each fixed $y$, the inner integral runs from $x = |y|$ to $x = 2$.
   • This describes a region where $y$ ranges from $-1$ to $1$, and $x$ starts at the absolute value of $y$ and goes up to $2$.

The Equality of Integrals:

The two integrals represent the same region in the $xy$-plane but are expressed with different bounds. The first integral expresses the region by integrating over $x$ first and then $y$, while the second integral reverses the order, integrating over $y$ first and then $x$. This is a typical application of Fubini's theorem, which states that if a function is integrable over a given region, the order of integration can be switched.

Questions for Further Exploration:

1. How does changing the order of integration affect the limits?
2. What are the conditions under which Fubini’s theorem allows switching the order of integration?
3. Can you find an explicit example function $f(x, y)$ and calculate both integrals?
4. How does the region of integration look graphically for both descriptions?
5. What happens if the limits of integration are modified slightly?

Tip: When switching the order of integration, sketching the region can help visualize and set up the correct bounds for the new order.