Find the area bounded by the curves !!y^2=x+1!! and !!x^2+y^2=1!!
Solution This question can be solved in more than one way let us look at those.
The area under a curve is considered as !!intxdy!! or !!intydx!!. In first case we consider the area under the curve as sum of the areas of a collection of horizontal rectangles, while the second case is for vertical rectangles.
The equation of the circle is !!x=+-sqrt(1-y^2)!! and equation of parabola is !!x=y^2-1!!
The shaded area between the circle and parabola is given by
!!=1/2 y sqrt(1-y^2) +1/2 tan^-1 y/sqrt(1-y^2) -y^3/3 +y|_-1^1!!
The shaded area can be considered as the area of the semicircle and the area enclosed between the parabola and the !!y!! axis.