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
$=2\underset{0}{\overset{3}{\int }}(\sqrt{9+y}-\sqrt{9-y})dy+2\underset{3}{\overset{9}{\int }}(\sqrt{9-y})dy$
$=2\left[\underset{0}{\overset{3}{\int }}(9+y{)}^{1/2}dy-\underset{0}{\overset{3}{\int }}(9-y{)}^{1/2}dy+\underset{3}{\overset{9}{\int }}(9-y{)}^{1/2}dy\right]$
$=2\left[\frac{2}{3}{\left[(9+y{)}^{3/2}\right]}_0^3+\frac{2}{3}{\left[(9-y{)}^{3/2}\right]}_0^3-\frac{2}{3}{\left[(9-y{)}^{3/2}\right]}_3^9\right]$
$=\frac{4}{3}[12\sqrt{12}-27+6\sqrt{6}-27-(0-6\sqrt{6})]$
$=\frac{4}{3}[24\sqrt{3}+12\sqrt{6}-54]$
$=8(4\sqrt{3}+2\sqrt{6}-9)$