To find the area of the region between the graph of the function y = 3x² + 2x and the x-axis from x = 1 to x = 3, we will first calculate the definite integral of the function over this interval.
The area A can be calculated using:
A = ∫[1 to 3] (3x² + 2x) dx
Now we will find the antiderivative of the function:
∫ (3x² + 2x) dx = x³ + x² + C
Next, we will evaluate this antiderivative from 1 to 3:
A = [ (3)³ + (3)² ] - [ (1)³ + (1)² ]
Calculating these values:
A = [27 + 9] - [1 + 1] = 36 - 2 = 34
Thus, the area of the region between the graph of y = 3x² + 2x and the x-axis from x = 1 to x = 3 is 34 square units.