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

The ellopse `(x^2)/(a^2)+y^(2)/(b^2)= 1` and the
line `x/a+y/b=1` can be drawn as shown in the given figure
Then , we have of the area of the shaded region
Required area
= {area between `x^2/a^2+y^2/b^2=1` and the x-axis from x=0 to x=a }
- {area between the line `x/a+y/b=1` and the x-axis from x=0 to x =a }
`=underset(0)overset(a)int"(y of the ellipse)"dx-underset(0)overset(a)int"(y of the line x)" dx `
`underset(0)overset(a)intb/a.sqrt(a^2-x2)dx-underset(0)overset(a)int(b(a-x))/(a)dx`
`=b/a.[(x(sqrt(a^2-x^2)))/2+a^2/2 sin^-1"""x/a]_0^a-b/a.[ax-x^2/2]_0^a`
`=(ab)/a(sin^(-1)1-sin^(-1)0)-(ab -(ab)/(2))=((piab)/4-(ab)/2)`sq units
Hence , the required area is `((piab)/4-(ab)/2)`sq units.