Area Under the Curve
Area with Inequality Constraint
nta_pyq_2024_jan
Grade 12
Question:
The area of the region $\left\{(x,y): y^2\le4x,\, x<4,\, \frac{xy(x-1)(x-2)}{(x-3)(x-4)}>0,\, x\ne3\right\}$ is
$\frac{16}{3}$
$\frac{64}{3}$
$\frac{8}{3}$
$\frac{32}{3}$
Step-by-Step Solution
Key Concept: Analyze the sign of $\frac{xy(x-1)(x-2)}{(x-3)(x-4)}$. Combined with $y^2\le4x$ (requiring $x\ge0$) and $x<4$: case $y>0$ gives $x\in(0,1)\cup(2,3)$; case $y<0$ gives $x\in(1,2)\cup(3,4)$. By symmetry, total area $=2\int_0^4\sqrt{x}\,dx$.
For $y>0$: $x\in(0,1)\cup(2,3)$. For $y<0$: $x\in(1,2)\cup(3,4)$. Together both half-parabolas are covered for all $x\in(0,4)$. Area $=2\int_0^4\sqrt{x}\,dx=2\cdot\frac{2}{3}\cdot8=\frac{32}{3}$.
Correct Answer: 4