Area under a curve

Question

Find the area bounded by the curves !!y^2=x+1!! and !!x^2+y^2=1!!

  • !!pi-4/3!!
  • !!pi-2/3!!
  • !!pi/2+4/3!!
  • None

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.

First approach:

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

!!int_(-1)^1 sqrt(1-y^2)-(y^2-1)dy!!

!!=1/2 y sqrt(1-y^2) +1/2 tan^-1 y/sqrt(1-y^2) -y^3/3 +y|_-1^1!!

!!=pi/2+2-(1/3+1/3)=pi/2+4/3!!

Second approach:
The shaded area can be considered as the area of the semicircle and the area enclosed between the parabola and the !!y!! axis.

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