question_answer

The area bounded by the curve \[x=4-{{y}^{2}}\]and the Y-axis is

A)  \[16\text{ }sq\text{ }unit\]

B)  \[32\text{ }sq\text{ }unit\]

C)  \[\frac{32}{3}\text{ }sq\text{ }unit~~\]  

D)  \[\frac{16}{3}sq\text{ }unit\]

Correct Answer: C

Solution :

Given, curve \[x=4-{{y}^{2}}\] and y-axis. The formed area is ABCOA                 \[=2\times \]area of ABOA                 \[=2\times \int_{0}^{4}{\sqrt{4-x}}dx\]                 \[=2\left[ \frac{{{(4-x)}^{3/2}}}{3/2} \right]_{0}^{4}\]                 \[=2\left[ -\frac{2}{3}\times 0+\frac{2}{3}{{(4)}^{3/2}} \right]\]                 \[=\frac{4}{3}\times 8=\frac{32}{3}sq\,unit\]