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.