This problem requires you to prove a specific equality involving double integrals. The given equation is

$\int_a^b \left( \int_a^s f(s,t) \, dt \right) ds = \int_a^b \left( \int_t^b f(s,t) \, ds \right) dt,$ where $a > 0$, $b > a$, and $f(s,t)$ is a continuous function.

Explanation:

This equation demonstrates a case of Fubini's Theorem or Tonelli's Theorem, which allows the interchange of the order of integration when integrating over certain types of regions.

Step 1: Interpretation of the Domains

1. On the left-hand side, the order of integration is $t$ first (from $a$ to $s$), then $s$ (from $a$ to $b$).

2. On the right-hand side, the order of integration is reversed, with $s$ first (from $t$ to $b$) and then $t$ (from $a$ to $b$).

The key idea is to recognize that both sides represent integrals over the same triangular region in the $(s,t)$-plane:

$\{(s,t) \mid a \leq t \leq s \leq b\}.$

On the left-hand side, we integrate first along lines parallel to the $t$-axis (keeping $s$ fixed), while on the right-hand side, we integrate first along lines parallel to the $s$-axis (keeping $t$ fixed).

Step 2: Change of Variables and Region of Integration

We can express the region of integration in both forms:

• In the first case, for fixed $s$, $t$ ranges from $a$ to $s$.
• In the second case, for fixed $t$, $s$ ranges from $t$ to $b$.

By switching the order of integration, you are effectively rearranging how you traverse the same triangular region. This allows you to apply Fubini's Theorem, which guarantees the equality of iterated integrals over the same region.

Step 3: Conclusion

Thus, by recognizing that the region of integration is the same in both cases and that the function $f(s,t)$ is continuous (a requirement for applying Fubini’s Theorem), we can conclude that the integrals are equal.

$\int_a^b \left( \int_a^s f(s,t) \, dt \right) ds = \int_a^b \left( \int_t^b f(s,t) \, ds \right) dt.$

Do you want more details or have any questions?

Here are some relative questions for further exploration:

1. How does Fubini’s Theorem apply to more general regions?
2. What are the requirements for switching the order of integration?
3. Can this result be extended to non-rectangular regions?
4. What happens if the function $f(s,t)$ is not continuous?
5. How would this proof change if $a$ or $b$ were infinite?

Tip: When dealing with double integrals, sketching the region of integration can greatly help in understanding the limits of the iterated integrals.