Question

Area lying in first quadrant and bounded by the circle x² + y² = 4 and the lines x = 0 and x = 2, is
(a) π

(b) $\frac{\mathrm{\pi }}{2}$

(c) $\frac{\mathrm{\pi }}{3}$

(d) $\frac{\mathrm{\pi }}{4}$

Answer 1

(a) π



x² + y² = 4 represents a circle with centre at origin O(0, 0) and radius 2 units, cutting the coordinate axis at A, A', B and B'.
x = 2 represents a straight line parallel to the y-axis, intersecting the circle at A(2, 0)
x = 0 represents the y-axis
Area bounded by the circle and the two given lines in the first quadrant is the shaded area OBCAO
$$\begin{gathered} \mathrm{Area}\left(OBCAO\right)={\int }_0^2\left|y\right|dx \\ ={\int }_0^2\sqrt{4-x^2}dx \\ ={\left[\frac{1}{2}x\sqrt{4-x^2}+\frac{1}{2}\times 4{\sin }^{-1}\left(\frac{x}{2}\right)\right]}_0^2 \\ =\frac{1}{2}\times 2\sqrt{4-2^2}+\frac{1}{2}\times 4{\sin }^{-1}\left(\frac{2}{2}\right)-0 \\ =0+2{\sin }^{-1}\left(1\right) \\ =2\times \frac{\mathrm{\pi }}{2} \\ =\mathrm{\pi }\mathrm{sq}\mathrm{units} \end{gathered}$$