Question

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

विकल्प

• π

• \[\frac{\pi}{2}\]
• \[\frac{\pi}{3}\]
• \[\frac{\pi}{4}\]

Answer 1

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
\[\text{ Area }\left( OBCAO \right) = \int_0^2 \left| y \right| dx\]
\[ = \int_0^2 \sqrt{4 - x^2} dx\]
\[ = \left[ \frac{1}{2}x\sqrt{4 - x^2} + \frac{1}{2} \times 4 \sin^{- 1} \left( \frac{x}{2} \right) \right]_0^2 \]
\[ = \frac{1}{2} \times 2\sqrt{4 - 2^2} + \frac{1}{2} \times 4 \sin^{- 1} \left( \frac{2}{2} \right) - 0\]
\[ = 0 + 2 \sin^{- 1} \left( 1 \right)\]
\[ = 2 \times \frac{\pi}{2}\]
\[ = \pi\text{ sq units }\]