Using integration, find the area in the first quadrant bounded by the curve y = x|x|, the circle x² + y² = 2and the y-axis.

y = x|x|

Now |x|, behaves differently when x < 0 and x > 0 but we are not concerned for x < 0 because we want to find the area in first quadrant

|x| = x for x > 0

Hence y = x² for x > 0

x² + y² = 2 is a circle with center (0, 0) and radius √2

Let us find where does the parabola and circle intersect at

For this solve the equation of parabola and circle

Put y = x² in equation of circle

⇒ x² + x⁴ = 2

By observation x = 1 and x = -1 satisfies the equation but we want to find area in first quadrant hence x = 1

Now put x = 1 in parabola equation to get the y coordinate

⇒ y = 1²

⇒ y = 1

Hence the parabola and circle intersect at (1, 1) in first quadrant

Roughly plot both the graphs y = x² which is a parabola and the circle x² + y² = 2 and the area required is as shown

We have to find the shaded region OAB

Construct a line x = 1 and the area OAB will be the area under the circular arc BA minus the area under the parabola arc OA

First let us write the equations of circle and parabola in the from y = f(x)

Circle: x² + y² = 2

Parabola: y = x²

The area is given by

Area OAB = area under circle curve AB – area under parabola curve OA

The integral is of the form where a = √2

We know that

Hence

Hence area is square units