Area Under the Curve
Area Under max{sin x, cos x} — A + A²
nta_pyq_2026_jan
Grade 12
Question:
Let the area of the region bounded by the curve $y=\max\{\sin x,\cos x\}$, lines $x=0$, $x=\dfrac{3\pi}{2}$, and the $x$-axis be $A$. Then $A+A^2$ is equal to _____
Step-by-Step Solution
Key Concept: Split at the crossover points of $\sin x$ and $\cos x$ on $[0,3\pi/2]$: at $x=\pi/4$ ($\cos$ leads) and $x=5\pi/4$ ($\sin$ leads). Integrate $\max\{\sin x,\cos x\}$ piecewise, taking absolute values below the x-axis.
$A=3$. $A+A^2=12$.
Correct Answer: 12