Question

Find the centroid of area enclosed by the parabola y²=ax and ay =x².​

Answer 1

The given curves are y=x

2

and y

2

=x i.e. y=

x

To find the intersection point of two curves, let us have

x

2

=

x

⟹x

4

=x

⟹x

4

−x=0

⟹x(x

3

−1)=0

⟹x=0 and x=1

⟹(0,0) and (1,1) are the points of intersection of two curves.

∴ Area bounded by these two curves =∫

0

1

(x

2

−

x

)dx

=

∣

∣

∣

∣

∣

∣

3

x

3

−

2

3

(x)

2

3

∣

∣

∣

∣

∣

∣

0

1

=

3

1

−

3

2

=−

3

1

Hence, area bounded by two parabolas y=x

2

and y

2

=x is −

3

1