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Question: Find the whole area of circle \(x ^ { 2 } + y ^ { 2 } = a ^ { 2 }\)...

Find the whole area of circle x2+y2=a2x ^ { 2 } + y ^ { 2 } = a ^ { 2 }

A

π\pi

B

πa2\pi a ^ { 2 }

C

πa3\pi a ^ { 3 }

D

a2a ^ { 2 }

Answer

πa2\pi a ^ { 2 }

Explanation

Solution

The required area is symmetric about both the axis as shown in figure

∴ Required area = 4∫0aa2−x2dx\int _ { 0 } ^ { a } \sqrt { a ^ { 2 } - x ^ { 2 } } d x =

4[x2a2−x2+a22sin⁡−1xa]0a4 \left[ \frac { x } { 2 } \sqrt { a ^ { 2 } - x ^ { 2 } } + \frac { a ^ { 2 } } { 2 } \sin ^ { - 1 } \frac { x } { a } \right] _ { 0 } ^ { a }

=4[π2×a22]=πa24 \left[ \frac { \pi } { 2 } \times \frac { a ^ { 2 } } { 2 } \right] = \pi a ^ { 2 }