Area Under the Curve
Area Between Curves — Circular and Parabolic Regions
nta_pyq_2023_apr
Grade 12
Question:
If the area of the region $S=\{(x,y):2y-y^2\leq x^2\leq 2y,\ x\geq y\}$ is equal to $\dfrac{n+2}{n+1}-\dfrac{\pi}{n-1}$, then the natural number $n$ is equal to _______.
Step-by-Step Solution
Key Concept: The boundaries are $x^2+y^2-2y=0$ (circle of radius 1 centred at $(0,1)$), $x^2=2y$ (upward parabola), and $x=y$. Find intersections: circle $\cap$ $x=y$ at $(1,1)$; parabola $\cap$ $x=y$ at $(2,2)$.
$A=\int_0^1\!\left(1+\sqrt{1-x^2}-\frac{x^2}{2}\right)dx+\int_1^2\!\left(x-\frac{x^2}{2}\right)dx=\frac{7}{6}-\frac{\pi}{4}=\frac{n+2}{n+1}-\frac{\pi}{n-1}\Rightarrow n=5$.
Correct Answer: 5