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