Area Under the Curve
Area in Region bounded by Circle and Parabola
nta_pyq_2023_jan
Grade None
Question:
Let $\Delta$ be the area of the region $\{(x,y)\in\mathbb{R}^2:x^2+y^2\leq21,\,y^2\leq4x,\,x\geq1\}$. Then $\dfrac{1}{2}\!\left(\Delta-21\sin^{-1}\dfrac{2}{\sqrt{7}}\right)$ is equal to:
$2\sqrt{3}-\dfrac{1}{3}$
$\sqrt{3}-\dfrac{2}{3}$
$2\sqrt{3}-\dfrac{2}{3}$
$\sqrt{3}-\dfrac{4}{3}$
Step-by-Step Solution
Key Concept: Region: inside circle $x^2+y^2=21$, right of $x=1$, inside parabola $y^2=4x$. $\Delta=2\int_1^3 2\sqrt{x}\,dx+2\int_3^{\sqrt{21}}\sqrt{21-x^2}\,dx$.
$\dfrac{1}{2}\!\left(\Delta-21\sin^{-1}\dfrac{2}{\sqrt{7}}\right)=\sqrt{3}-\dfrac{4}{3}$.
Correct Answer: 4