What is the area of the region bounded by the curve y=2x^2+2x, x axis and lines x=1 and x=2?

We'll have to determine the definite integral of the given
function. We'll use Leibniz-Newton formula to calculate the definite
integral:

Int f(x)dx =
F(b)-F(a)

We'll calculate the indefinite integral of
f(x):

Int f(x)dx = Int (2x^2 +
2x)dx

We'll use the property of integral to
be additive:

Int (2x^2 + 2x)dx = Int 2x^2dx + Int
2xdx

Int 2x^2dx = 2*x^3/3 +
C

Int 2xdx = 2*x^2/2 + C

We'll
reduce and we'll get:

Int 2xdx = x^2 +
C

Int (2x^2 + 2x)dx = 2x^3/3 + x^2 +
C

F(2) - F(1) = 2^4/3 + 2^2 - 2/3 -
1^2

F(2) - F(1) = 14/3 +
3

F(2) - F(1) =
23/3

The area bounded by the curve of f(x), x
axis and the lines x=1, x=2  is A=23/3 square
units.