Example 1 - Chapter 8 Class 12 Application of Integrals

Last updated at April 16, 2024 by Teachoo

Transcript

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

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, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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