Find the area of the smaller region bounded by the ellipse `(x^2)/(a^2)+(y^2)/(b^2)=1`and the line `x/a+y/b=1`

Explanation

`(x^(2))/(a^(2))+(y^(2))/(b^(2))=1`
`y=(b)/(a)sqrt(a^(2)-x^(2)) " " ` ...(1)
`(x)/(a)+(y)/(b)=1impliesy=(b)/(a)(a-x) " "` ...(2)
The points of intersection of the ellipse `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1` and straight line `(x)/(a)+(y)/(b)=1` are `A(a, 0) and B(0, b)`.
Draw the graphs of both.

Now required area
`=ar(OAPBO)-ar(OAQBO)`
`=int_(0)^(a)(b)/(a)sqrt(a^(2)-x^(2))dx-int_(0)^(a)(b)/(a)(a-x)dx`
`=(b)/(a)[(x)/(2) sqrt(a^(2)-x^(2))+(a^(2))/(2)"sin"^(-1)(x)/(a)]_(0)^(a)-(b)/(a)[ax-(x^(2))/(2)]_(0)^(a)`
`=(b)/(a)[(0+(a^(2))/(2)*(pi)/(2))-(0+0)]-(b)/(a)[(a^(2)-(a^(2))/(2))-(0-0)]`
`=((pi ab)/(4)-(ab)/(2))` sq. units.