Area Under the Curve
Area of Region Defined by Inequalities
nta_pyq_2024_jan
Grade 12
Question:
Let the area of the region $\{(x,y): x-2y+4\ge0,\, x+2y^2\ge0,\, x+4y^2\le8,\, y\ge0\}$ be $\frac{m}{n}$, where $m$ and $n$ are coprime numbers. Then $m+n$ is equal to
Step-by-Step Solution
Key Concept: Identify the region bounded by the parabolas $x=-2y^2$ and $x=8-4y^2$, and line $x=2y-4$. Find intersection points, split the region and integrate w.r.t. $y$.
Integrating w.r.t. $y$ from 0 to $3/2$ (with appropriate split at $y=1$):
$A=\int_0^1[(8-4y^2)-(-2y^2)]dy+\int_1^{3/2}[(8-4y^2)-(2y-4)]dy=\frac{107}{12}$.
$m+n=107+12=119$.
Correct Answer: 119