Question

If $f_1$ and $f_2$ are integrable on the region $R$ in the plane $\mathbb{R}^2$ and if $f_1(x, y) \leq f_2(x, y)$ for all $(x, y)$ in $R$ then which of the following always hold:

• A. $\iint_R f_1(x, y) \, dx dy = \iint_R f_2(x, y) \, dx dy$
• B. $\iint_R f_2(x, y) \, dx dy \leq \iint_R f_1(x, y) \, dx dy$
• C. $\iint_R f_1(x, y) \, dx dy < \iint_R f_2(x, y) \, dx dy$
• D. $\iint_R f_1(x, y) \, dx dy \leq \iint_R f_2(x, y) \, dx dy$

Hint:

Integration preserves inequalities! If $f \leq g$, then $\int f \leq \int g$. Always remember that equality can hold if $f(x, y) = g(x, y)$ on $R$.

Answer 1

The Correct Option is D

Step 1: Concept
This question tests the Monotonicity Property (Order Preservation) of double integrals over bounded regions in $\mathbb{R}^2$.

Step 2: Key Formulas and Approach
If $g(x, y) \geq 0$ for all $(x, y) \in R$, then by definition of the Riemann integral (as a limit of non-negative Riemann sums): \[ \iint_R g(x, y) \, dx dy \geq 0 \] Letting $g(x, y) = f_2(x, y) - f_1(x, y) \geq 0$, linearity of integration yields the monotonic property.

Step 3: Step-by-step Explanation

• Given $f_1(x, y) \leq f_2(x, y)$ for all $(x, y) \in R$.
• Subtract $f_1(x, y)$ from both sides: \[ f_2(x, y) - f_1(x, y) \geq 0 \quad \forall (x, y) \in R \]
• Integrate both sides over the region $R$: \[ \iint_R \left[ f_2(x, y) - f_1(x, y) \right] dx dy \geq 0 \]
• By linearity of double integrals: \[ \iint_R f_2(x, y) \, dx dy - \iint_R f_1(x, y) \, dx dy \geq 0 \]
• Rearranging terms: \[ \iint_R f_1(x, y) \, dx dy \leq \iint_R f_2(x, y) \, dx dy \]
• Note that strict inequality Option (C) is false because if $f_1 = f_2$, their integrals are equal.

Step 4: Final Answer
The monotonicity property guarantees that $\iint_R f_1(x, y) \, dx dy \leq \iint_R f_2(x, y) \, dx dy$. Thus, Option (D) is correct.