Question

In the adjoining, OABO is the region of the ellips `9x^(2)+y^(2)=36` which lies in first quadrant. If `OA= 2,OB=6,` then find the area of the region bounded by chord AB and arc AB.

Answer 1

Equation of AB
`y-0=(6-0)/(0-2)(x-2)`
`impliesy= -3x+6`
Required area
`=int_(0)^(2)3sqrt(4-x^(2))dx-int_(0)^(2)(-3x+6)dx`
`=3[(x)/(2)sqrt(4-x^(2))+(4)/(2)"sin"^(-1)(x)/(2)]_(0)^(2)-[(-3x^(2))/(2)+6x]_(0)^(2)`
` = 3[(0+2"sin"^(-1)1)-(0+0)]-[(-6+12)-(0+0)]`
`=(3pi-6)sq.` units.