Find the area between y= x and f(x) = x^2

Given the line  y= x and the parabola f(x)=
x^2.

To find the area between the line and the parabola, we need to
determine the intersection points between the line and the curve.

To
find the intersection points, we will caculate x values such that y =
f(x).

==> x^2 = x

==> x^2
- x = 0

==> x ( x-1) =
0

==> x= 0 , 1

Now we will
determine the area under the line y= x between x= 0 and x=
1.

==> A1 = integral
y

              = intg x 
dx

              = x^ 2/2 

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

Now we will calculate the area under the
curve f(x) = x^2 and x= 0 and x= 1.

==> A2 = intg f(x)
dx

              = intg x^2
dx

                = x^3 /3 +
C

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

Then,
the area between y and f(x) is:

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

Then the area = 1/6 square
units.