CBSE Class 12 Mathematics Applications of Integrals Q #1710
COMPETENCY BASED
APPLY
1 Marks 2026 AISSCE(Board Exam) MCQ SINGLE
Which of the following expressions will give the area of region bounded by the curve $y=x^{2}$ and line $y=16$?
(A) $\int_{0}^{4}x^{2}dx$
(B) $2\int_{0}^{4}x^{2}dx$
(C) $\int_{0}^{16}\sqrt{y}dy$
(D) $2\int_{0}^{16}\sqrt{y}dy$
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Correct Answer: D

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Detailed Solution

Step 1: Visualize the Region

The curve is a parabola $y = x^2$ opening upwards with vertex at $(0,0)$. The line $y = 16$ is a horizontal line intersecting the parabola at points where $x^2 = 16$, i.e., $x = -4$ and $x = 4$. The region is symmetric about the y-axis.

Step 2: Formulate using y-limits

To find the area using integration with respect to $y$, we express $x$ in terms of $y$: $x = \pm\sqrt{y}$. The width of the region at any height $y$ is the distance between the two branches of the parabola, which is $\sqrt{y} - (-\sqrt{y}) = 2\sqrt{y}$. The limits for $y$ are from $0$ to $16$. Thus, the area is: $$Area = \int_{0}^{16} 2\sqrt{y} dy$$

Step 3: Verify with Options

Comparing our derived expression $2\int_{0}^{16}\sqrt{y}dy$ with the given options, it matches option (D).

Final Answer: D

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student must translate a geometric description into a definite integral using the concept of area between curves.
Knowledge Dimension: PROCEDURAL
Justification: The student is required to execute the algorithm of setting up an integral based on the geometric properties of the function.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. This question tests the student's ability to choose the correct variable of integration and apply symmetry to simplify the calculation of area under a curve.

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