Question

The area of the region bounded by the line $y=4$ and the curve $y=x^{2}$ is ________.

• A. $\frac{32}{3}$ square units
• B. 0 square unit
• C. 1 square unit
• D. 32 square units

Hint:

Area $= 2 \times \int_{0}^{2} (4-x^2) dx$ due to symmetry.

Answer 1

The Correct Option is A

Step 1: Concept
Area between curves: $\int (y_{upper} - y_{lower}) dx$.
Step 2: Analysis
Points of intersection: $x^2 = 4 \Rightarrow x = \pm 2$.
Area $= \int_{-2}^{2} (4 - x^2) dx$.
Step 3: Calculation
Area $= [4x - x^3/3]_{-2}^{2}$
$= (8 - 8/3) - (-8 + 8/3) = (16/3) - (-16/3) = 32/3$.
Step 4: Conclusion
The area is $\frac{32}{3}$ square units.
Final Answer:(A)