Question

What is the area bounded by the curve $x=4-y^2$ and the y-axis.

• A. $16$ sq units.
• B. $32$ sq units
• C. $\frac{32}{3}$ sq units
• D. $\frac{16}{3}$ sq units.

Answer 1

The correct option is C

$\frac{32}{3}$ sq units

Explanation of correct answer :

Finding the area bounded by the curves :

The graph for the curves is as shown:

The shaded portion is the area bounded by the curve $x=4-y^2$ and y-axis.

The required area is given as:

$$\begin{gathered} \Rightarrow \text{Area of}ABCOA \\ \Rightarrow 2\times \text{Area o}fABOA \\ \Rightarrow 2\times {\int }_0^4\sqrt{4-x}dx \\ \Rightarrow 2\times {\left[-\frac{{\left(4-x\right)}^{{\displaystyle \frac{3}{2}}}}{{\displaystyle \frac{3}{2}}}\right]}_0^4 \\ \Rightarrow 2\times \left[-\frac{2}{3}\times 0+\frac{2}{3}{\left(4\right)}^{\frac{3}{2}}\right] \\ \Rightarrow 4\times \frac{8}{3} \\ \Rightarrow \frac{32}{3} \end{gathered}$$

Thus, The area within the curve is $\frac{32}{3}$sq unit.

Hence, the correct answer is Option(C).