question_answer

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

A)  16 sq unit                           

B)  32 sq unit

C) \[\frac{32}{3}\]sq unit   

D)        \[\frac{16}{3}\]sq unit

Correct Answer: C

Solution :

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