Area of the region bounded by the curve $y=\sqrt{49-x^2}$ and $X$-axis is

• A

  $49 \pi$ sq. units
• Correct answer
  B

  $\frac{49 \pi}{2}$ sq. units
• C

  $\frac{49 \pi}{4}$ sq. units
• D

  $98 \pi$ sq. units

Solution

$\begin{aligned} \text { Required area } & =2 \int_0^7 \sqrt{49-x^2} \mathrm{~d} x \\ & =2\left[\frac{x}{2} \sqrt{49-x^2}+\frac{49}{2} \sin ^{-1}\left(\frac{x}{7}\right)\right]_0^7 \\ & =2\left[\frac{7}{2} \sqrt{49-49}+\frac{49}{2} \sin ^{-1}\left(\frac{7}{7}\right)\right]-0 \\ & =2 \times \frac{49}{2} \times \frac{\pi}{2} \\ & =\frac{49 \pi}{2} \text { sq. units } \end{aligned}$

Hence, the correct answer is option: 'B'.