Example 1 Find the area enclosed by the circle 𝑥2 + 𝑦2 = 𝑎2Given
𝑥^2 + 𝑦^2= 𝑎^2
This is a circle with
Center = (0, 0)
Radius = 𝑎
Since radius is a,
OA = OB = 𝑎
A = (𝑎, 0)
B = (0, 𝑎)
Now,
Area of circle = 4 × Area of Region OBAO
= 4 × ∫1_𝟎^𝒂▒〖𝒚 𝒅𝒙〗
Here,
y → Equation of Circle
We know that
𝑥^2 + 𝑦^2 = 𝑎^2
𝑦^2 = 𝑎^2− 𝑥^2
y = ± √(𝑎^2−𝑥^2 )
Since AOBA lies in 1st Quadrant
y = √(𝒂^𝟐−𝒙^𝟐 )
Now,
Area of circle = 4 × ∫1_0^𝑎▒〖𝑦 𝑑𝑥〗
= 4 × ∫1_0^𝑎▒〖√(𝑎^2−𝑥^2 ) 𝑑𝑥〗
Using: √(𝑎^2−𝑥^2 )dx = 1/2 √(𝑎^2−𝑥^2 ) + 𝑎^2/2 〖"sin" 〗^(−1) 𝑥/4 + c
= 4[𝒙/𝟐 √(𝒂^𝟐−𝒙^𝟐 )+𝒂^𝟐/𝟐 〖"sin" 〗^(−𝟏) 𝒙/𝒂]_𝟎^𝒂
= 4[𝑎/2 √(𝑎^2−𝑎^2 )+𝑎^2/2 〖"sin" 〗^(−1) 𝑎/𝑎−0/2 √(𝑎^2−0)−0^2/2 〖"sin" 〗^(−1) (0)]
= 4[0+𝑎^2/2 〖"sin" 〗^(−1) (1)−0−0]
= 4.𝑎^2/2. 𝜋/2
= 𝝅𝒂^𝟐

Made by

Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo

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