Area Under the Curve
Area with Hyperbola and Parabola Boundaries
nta_pyq_2026_jan
Grade 12
Question:
The area of the region $R=\{(x,y):xy\leq8,\,1\leq y\leq x^2,\,x\geq0\}$ is
$\dfrac{2}{3}(24\log_e 2-7)$
$\dfrac{1}{3}(40\log_e 2+27)$
$\dfrac{2}{3}(20\log_e 2+9)$
$\dfrac{1}{3}(49\log_e 2-15)$
Step-by-Step Solution
Key Concept: Intersection points: $y=1$ and $y=x^2$ at $x=1$; $xy=8$ and $y=x^2$ at $x=2$; $xy=8$ and $y=1$ at $x=8$. Region: $x\in[1,2]$ upper by $y=x^2$, lower by $y=1$; $x\in[2,8]$ upper by $y=8/x$, lower by $y=1$.
$A=\frac{2}{3}(24\log_e2-7)$.
Correct Answer: 1