To calculate the required area between f(x)=3x^2+2x and x=1 and x=2, we need the definite integral of f(x) between x = 1 and x = 2
Int[ 3x^2 + 2x dx], x =1 to x = 2
=> 3x^3/3 + 2x^2/2 , x = 1 to x = 2
=> x^3 + x^2, x = 1 to x = 2
=> 2^3 + 2^2 - 1^3 - 1^2
=> 8 + 4 - 1 - 1
The required area is 10 square units.
Since there is not established the other boundary curve, we'll suppose that we have to calculate the area between f(x), the lines x=1 and x=2 and the x axis.
The definite integral will be calculated with Leibniz-Newton formula:
Int f(x)dx = F(b)-F(a)
We'll calculate the indefinite integral of f(x):
Int f(x)dx = Int (3x^2 + 2x)dx
We'll use the property of integral to be additive:
Int (3x^2 + 2x)dx = Int 3x^2dx + Int 2xdx
Int 3x^2dx = 3*x^3/3 + C
Int 3x^2dx =x^3 + C
Int 2xdx = 2*x^2/2 + C
Int 2xdx = x^2 + C
Int (3x^2 + 2x)dx = x^3 + x^2 + C
F(2) - F(1) = 2^3 + 2^2 - 1^3 - 1^2
F(2) - F(1) = 8 + 4 - 2
F(2) - F(1) = 10
The area bounded by the curve of f(x) and the lines x=1, x=2 and x axis is A=10.