y=x and y = x^2

First we need to find the points of intersection that bounded area:

==> y = y

==> x^2 = x

==> x^2 - x = 0

==> x(x-1) = 0

Then x = 0 and x= 1

Then we will find the area between x^2 x = 0, and x= 1

==> We know that the area is:

A1 = intg y = intg x dx = x^2 /2

==> A1 = (1/2 - 0) = 1/2

== A2 = intg y= intg x^2 = x^3/3

==> A2 = ( 1/3- 0 ) = 1/3

Then the area is:

A = A1 - A2 = 1/2 - 1/3 = 1/6

**Then the area is 1/6 square units**

To find the area betwen y = x and x^2.

The intersection points of y = x and y = x^2 is given by:

x= x^2.

Or x-x^2 = 0.

Or x(1-x) = 0.

So x= 0 , Or 1-x = 0 , Or x = 1.

So the area between curves is to be found from x= 0 to x = 1.

If we draw the graph , y = x is above x = x^2 from x= 0 to x = 1 .

Therefore area between y = x and y =x^2 is given by:

Area = Integral (x-x^2)dx from x= 0 to x = 1.

Area = {(x^2/2 -x^3/3 at x= 1} - {(x^2/2 -x^3/3 at x= 0}

Area = { 1/2-1/3}- 0

Area = (3-2)/6 = 1/6.

Therefore the area between the curves = 1/6 sq units.