Class 12

Math

Calculus

Application of Integrals

Find the area of the smaller part of the circle $x_{2}+y_{2}=a_{2}$cut off by the line $x=2 a $

$=∫_{2a}2a_{2}−x_{2} dx$

$dx=acosθdθ$

when $x=2 a ,sinθ=2 1 $

$θ=4π $

when $x=a,sinθ=1$

$θ=2π $

$=2∫_{4π}(acosθ)(acosθ)dθ$

$=2a_{2}∫_{4π}(cos2θ+1)dθ$

$=a_{2}[[sin22θ ]+[θ]]$

$=a_{2}[[−21 ]+[4π ]]$

$=a_{2}[4π −21 ]$

answer