1. **Problem Statement:** We want to understand and write Fubini's Theorem for double integrals in an exam setting. 2. **Theorem Statement:** If $R$ is the rectangle defined by $a \leq x \leq b$ and $c \leq y \leq d$, and $f(x,y)$ is continuous on $R$, then the double integral over $R$ can be computed as an iterated integral in either order: $$\iint_R f(x,y) \, dA = \int_c^d \int_a^b f(x,y) \, dx \, dy = \int_a^b \int_c^d f(x,y) \, dy \, dx$$ 3. **Explanation:** - The double integral $\iint_R f(x,y) \, dA$ represents the volume under the surface $z = f(x,y)$ over the rectangle $R$. - Fubini's Theorem allows us to compute this double integral as an iterated integral, integrating first with respect to $x$ then $y$, or vice versa. - This is valid because $f$ is continuous on the rectangle $R$. 4. **How to write in exam:** - Clearly state the rectangle $R$ with bounds $a \leq x \leq b$, $c \leq y \leq d$. - State the continuity condition of $f(x,y)$ on $R$. - Write the equality of the double integral and the two iterated integrals as shown above. - Optionally, mention that this theorem justifies changing the order of integration. 5. **Summary:** Fubini's Theorem is a fundamental tool to evaluate double integrals by converting them into iterated integrals with respect to one variable at a time.