Mathematics - Area Under Curves Question with Solution | TestHub

The given curve is $y=2\sqrt{1-x^2}$.

Squaring both sides, we get $y^2=4(1-x^2)\Rightarrow x^2+\frac{y^2}{4}=1$.

This represents an ellipse with semi-minor axis $a=1$ and semi-major axis $b=2$.

Since $y\ge 0$, the given curve represents the upper half of the ellipse.

The area of the complete ellipse is $\pi ab=\pi (1)(2)=2\pi$.

Therefore, the area of the upper half of the ellipse is $\frac{1}{2}(2\pi )=\pi$.

Alternatively, the area can be found by integration:

Area $={\int }_{-1}^{1}2\sqrt{1-x^2}dx$

Area $=2{\left[\frac{x}{2}\sqrt{1-x^2}+\frac{1}{2}{\sin }^{-1}x\right]}_{-1}^1$

Area $=2\left[\frac{1}{2}\left(\frac{\pi }{2}\right)-\frac{1}{2}\left(-\frac{\pi }{2}\right)\right]=\pi$ sq. units.

Answer: $\pi$