Class 12

Math

Calculus

Application of Integrals

Find the area of the smaller region bounded by the ellipse $a_{2}x_{2} +b_{2}y_{2} =1$ and the line $ax +by =1$

$∴AreaBACB=Area(OBCAO)−Area(OBAO)$

$=∫_{0}b1−a_{2}x_{2} dx−∫_{0}b(1−ax )dx$

$=ab ∫_{0}a_{2}−x_{2} dx−ab ∫_{0}(a−x)dx$

$=ab [{2x a_{2}−x_{2} +2a_{2} sin_{−1}ax }_{0}−{ax−2x_{2} }_{0}]$

$=ab [{2a_{2} (2π )}−{a_{2}−2a_{2} }]$

$=ab [4a_{2}π −2a_{2} ]$

$=2aba_{2} [2π −1]$

$=2ab [2π −1]$

$=4ab (π−2)$