Question

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:

• A. A, B Only
• B. A, C, D, E Only
• C. A, C, D Only
• D. C, E Only

Hint:

Double integrals preserve basic properties of single integrals: linearity, domain additivity, and scaling by constants.

Answer 1

The Correct Option is B

Step 1 : Concept:
This question tests basic integral calculus properties, including single integrals for area under a curve and double integral properties over bounded regions.

Step 2 : Key Formulas and Approach:
1. Area under $y = f(x) \ge 0$ on $[a,b]$: $\text{Area} = \int_a^b f(x) dx$.
2. Linearity of double integrals: $\iint_R C f(x,y) dA = C \iint_R f(x,y) dA$.
3. Domain additivity: If $R = R_1 \cup R_2$ with disjoint interiors, $\iint_R f dA = \iint_{R_1} f dA + \iint_{R_2} f dA$.
4. Constant function integration: $\iint_R k \, dA = k \iint_R dA = k \times \text{Area}(R)$.

Step 3 : Step-by-step Explanation:

• Statement A:
By definition of the Riemann integral, the area under the curve of a non-negative continuous function $f(x) \ge 0$ on $[a,b]$ is given by $\int_a^b f(x) dx$. Statement A is correct.

• Statement B:
If $f(x,y) \ge 0$ on a region $D$ of non-zero area, $\iint_D f(x,y) dx dy$ must be greater than or equal to zero, and it is strictly positive unless $f(x,y) = 0$ almost everywhere. Stating it is generally $0$ is false. Statement B is incorrect.

• Statement C:
By the domain additivity property of multiple integrals, integrating over a region $R$ partitioned into non-overlapping subregions $R_1$ and $R_2$ equals the sum of integrals over $R_1$ and $R_2$. Statement C is correct.

• Statement D:
Factoring out the constant $k$: \[ \iint_R k \, dx dy = k \iint_R dx dy = k \times \text{Area}(R) \] Statement D is correct.

• Statement E:
By linearity of integration, constant multipliers can be factored outside the double integral sign. Statement E is correct.

Step 4 : Final Answer:
Statements A, C, D, and E are correct. Therefore, option (B) is the correct answer.