Question

find the area of the region bounded by the lines x=2 and x=-2 and the circle whose centre is 0,0 and radius 3 cm​

Answer 1

Given :  region bounded by the lines x=2 and x=-2 and the circle whose centre is 0,0 and radius 3 cm​

To Find :   area of the region

Solution:

circle whose centre is 0,0 and radius 3 cm​

circle is x² + y² = 3²

=> y² = 3² - x²

=> y  = √(3² - x²)

x=2 and x=-2

area of the region bounded by the lines x=2 and x=-2 and the circle

$2 \times \int\limits^{-2}_2 {\sqrt{3^2-x^2} } \, dx$

above and below x axis that's why multiplied by 2

∫√(a² - x²)dx  =  (x/2)√(a² - x²)  + (a²/2)Sin⁻¹)(x/a) + c

= 2 *   (x/2)√(3² - x²)  + (a²/2)Sin⁻¹)(x/a)  $]_{-2}^2$

= 2 *[ (2/2)√(3² - 2²)  + (3²/2)Sin⁻¹(2/3)  - ( (-2/2)√(3² - (-2)²)  + (3²/2)Sin⁻¹(-2/3) )]

= 2 & [√5  +  (9/2)Sin⁻¹(2/3) -  (-√5  -  (9/2)Sin⁻¹(2/3))]

= 2 *[ (2√5 + 9/2)Sin⁻¹(2/3) +  (9/2)Sin⁻¹)(2/3) ]

= 2 * (2√5 + 9 Sin⁻¹(2/3))

=  4√5  + 18 * 0.7297

= 8.944 + 13.135

= 22.079  cm²

area of the region bounded = 22.079  cm²

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