Calculate the area enclosed by y = X^2 and y = x + 2.

y= x^2

y=
x+2

To find the area will need to find the integral for both
equations:

First we need to determine the intersecting
points:

==> x^2 = x+
2

==> x^2 -x -2 = 0

==>
(x-2)(x+1) = 0

==> x= 2   x =
-1

Then we need to find the integral from x= -1 to x=
2

First let us find the area under y= x^2 from x= -1 to
2

==> A1 = intg ( x^2 )
dx

           = x^3/3

            = (
2^3/3) - (-1)^3 / 3

            = ( 8 + 1)/3 = 9/3 =
3

Then the area under y=x^2 from (-1 to 2 ) is  3 square
units>

Now we will calculate the area under y=
x+2:

A2 = intg (x+ 2)

    = x^2/2 +
2x

       = ( 2^2/2 + 2*2) - ( -1^2/2 +
2*-1)

        = 2 + 4) - ( 1/2 -
2)

       = 6 + 3/2 = 15/2 = 7.5 square
units:

Then the area is:

A = A2 - A1 =
7.5 - 3 = 4.5

Then, the area between y= x^2 and y=
x+2  is 4.5 square units