Area Under the Curve
Area Between Parabolas with Additional Constraint
nta_pyq_2023_apr
Grade 12
Question:
The area of the region $\{(x,y):\ x^2\leq y\leq 8-x^2,\ y\leq 7\}$ is
Step-by-Step Solution
Key Concept: The two parabolas $y=x^2$ and $y=8-x^2$ meet at $(\pm 2,4)$. The constraint $y\leq 7$ cuts off $y=8-x^2$ at $x=\pm 1$. Split the area symmetrically about the $y$-axis.
$A=2\left[\frac{2}{3}y^{3/2}\Big|_0^4+\left(-\frac{2}{3}(8-y)^{3/2}\right)\Big|_4^7\right]=\frac{4}{3}(8+8-1)=20$ sq. units.
Correct Answer: 3