Find the area bounded by the curve f(x) = x^2 and the line y= 5x-6.

Given the curve f(x) = x^2 and the line y=
5x-6

We need to find the area between the curve and the
line.

First we need to find the intersection
points.

==> x^2 =
5x-6

==> x^2 - 5x + 6 =
0

==> (x-2)(x-3) = 0

==>
x = 2 ==> x = 3

Then we need to find the area between the
interval [ 2, 3].

First we will find the ares under the curve f(x) =
x^2.

==> Int f(x) = Int x^2 dx =
x^3/3

==> A1 = F(3) - F(2) = 27/3 - 8/3 =
19/3.

Then the area under the curve f(x) is A1=
19/3.

Now we will calculate the area under the line
y.

==> Int y = Int 5x -6 dx = 5x^2/2 -
6x

==> Y(3) = (5/2)*9 - 18 = (45-36)/2 =
9/2

==> Y(2) = (5/2)*4 - 12 = 10 - 12 =
-2

==> y(3) - y(2) = 4.5 + 2 = 6.5 =
13/2

Then the area between the line and the curve is
:

A = A2 - A1 = 13/2 - 19/3 =
0.166666

Then the ares is given by 0.16666 square
units.