Area Under Curves
Area between piecewise linear and semicircle
MJAT_TS2_P1
Grade 12
Question:
Let $f(x)=\min\{|x|+1,\,3-|x|\}$ and $g(x)=\sqrt{9-x^2}$. If $S$ is the region in the $xy$-plane defined by $S=\{(x,y): f(x)\leq y\leq g(x)\}$, then the area of $S$ is:
A) $\dfrac{9}{2}\pi - 6$
B) $\dfrac{9}{4}\pi - \dfrac{7}{2}$
C) $\dfrac{9}{2}\pi - 7$
D) $9\pi - 14$
Step-by-Step Solution
Key Concept: Area of $S$ = Area under semicircle $g(x)$ minus area under $f(x)$, both over $[-3,3]$. Semicircle area $= \frac{\pi(3)^2}{2} = \frac{9\pi}{2}$. For $f(x)$: on $[0,1]$ it's $x+1$ (trapezoid, area $1.5$); on $[1,3]$ it's $3-x$ (triangle, area $2$). Total area under $f$ on $[-3,3]$ is $2\times 3.5 = 7$.
Area$(S) = \frac{9\pi}{2} - 7$.
Correct Answer: C