Area Under the Curve
Area Bounded by Parabola, Line and Circle (excluded region)
nta_pyq_2023_apr
Grade 12
Question:
If the area bounded by the curve $2y^2=3x$, lines $x+y=3$, $y=0$ and outside the circle $(x-3)^2+y^2=2$ is $A$, then $4(\pi+4A)$ is equal to __________.
Step-by-Step Solution
Key Concept: Integrate the area between $x=\frac{2y^2}{3}$ and $x=3-y$ from $y=0$ to $y=\frac{3}{2}$ (intersection), then subtract the $45°$ sector of the circle $(x-3)^2+y^2=2$ that lies in the first quadrant region.
$A=\frac{21}{8}-\frac{\pi}{4}$. $\pi+4A=\pi+\frac{21}{2}-\pi=\frac{21}{2}$. $4(\pi+4A)=42$.
Correct Answer: 42