Area Under the Curve
Area of Region — Bounded by Two Curves
nta_pyq_2026_jan
Grade 12
Question:
If the area of the region $\{(x,y):1-2x\leq y\leq4-x^2,\,x\geq0,\,y\geq0\}$ is $\dfrac{\alpha}{\beta}$, $\alpha,\beta\in\mathbf{N}$, $\gcd(\alpha,\beta)=1$, then the value of $(\alpha+\beta)$ is:
Step-by-Step Solution
Key Concept: The curves $y=1-2x$ and $y=4-x^2$ intersect at $x=1/2$ (where $y=0$ is also relevant). Boundaries: $y=0$ (x-axis), $y=4-x^2$ (parabola), $y=1-2x$ (line, zero at $x=1/2$). Region is from $x=0$ to $x=2$ under the parabola, accounting for the line.
Area $=61/12$. $\alpha+\beta=73$.
Correct Answer: 3