List of top Mathematics Questions on Double and triple integrals

Which of the following statements are correct:
A. Area of a non negative continuous function from $x = a$ to $x = b$ is $\int_a^b f(x) dx$.
B. $\iint_D f(x,y) dx dy = 0$, where $f(x,y) \ge 0 \, \forall (x,y)$ in $D$.
C. $\iint_R f(x,y) dx dy = \iint_{R_1} f(x,y) dx dy + \iint_{R_2} f(x,y) dx dy$, where $R$ is divided into $R_1$ and $R_2$, and $f$ is integrable on $R_1$ and $R_2$. D. If $f(x,y) = k \, \forall (x,y)$ in $R$, then $\iint_R f(x,y) dx dy = k \times (\text{area of } R)$.
E. $\iint_R C f(x,y) dx dy = C \iint_R f(x,y) dx dy$; $C$ is a constant
Choose the correct answer from the options given below:

The value of $\iiint_B (x + 2y + 3z)^2 \, dx \, dy \, dz$ is, where $B$ is the box $[0, 1] \times \left[-\frac{1}{2}, 0\right] \times \left[0, \frac{1}{3}\right]$

The value of $\iint_S \cos(x) \sin(y) \, dx \, dy$ is, where $S$ is $\left[0, \frac{\pi}{2}\right] \times \left[0, \frac{\pi}{2}\right]$

Let $f(x, y) = x^2 + y^2$ and $R = [-1, 1] \times [0, 1]$, then $\iint_R f(x, y) \, dx \, dy$ is

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: