Question

The area bounded by the curve $y=\left|x^{2}-9\right|$ and the line $y =3$ is:

• A. $4(2 \sqrt{3}+\sqrt{6}-4)$
• B. $4(4 \sqrt{3}+\sqrt{6}-4)$
• C. $8(4 \sqrt{3}+3 \sqrt{6}-9)$
• D. none

Answer 1

Answer: (D)

Area of shaded region
$=2 \int\limits_{0}^{3}(\sqrt{9+y}-\sqrt{9-y}) d y+2 \int\limits_{3}^{9}(\sqrt{9-y}) d y$
$=2\left[\int\limits_{0}^{3}(9+y)^{1 / 2} d y-\int\limits_{0}^{3}(9-y)^{1 / 2} d y+\int\limits_{3}^{9}(9-y)^{1 / 2} d y\right] $
$=2\left[\frac{2}{3}\left[(9+y)^{3 / 2}\right]_{0}^{3}+\frac{2}{3}\left[(9-y)^{3 / 2}\right]_{0}^{3}-\frac{2}{3}\left[(9-y)^{3 / 2}\right]_{3}^{9}\right]$
$=\frac{4}{3}[12 \sqrt{12}-27+6 \sqrt{6}-27-(0-6 \sqrt{6})]$
$=\frac{4}{3}[24 \sqrt{3}+12 \sqrt{6}-54]$
$=8(4 \sqrt{3}+2 \sqrt{6}-9)$