Final answer:
To find the surface area bounded by the curves y = -x³ + x² + 2x and the x-axis, we can integrate the absolute value of the derivative of the curve between the x-intercepts.
Step-by-step explanation:
The surface area bounded by the curves y = -x³ + x² + 2x and the x-axis can be found by integrating the expression for the curve from the lower limit to the upper limit. The x-values where the curve intersects the x-axis can be found by setting y = 0 and solving for x. In this case, setting -x³ + x² + 2x = 0 gives us two x-values -2 and 0.
To find the surface area, we can integrate the absolute value of the derivative of the curve between -2 and 0. The derivative of -x³ + x² + 2x is -3x² + 2x + 2. Taking the absolute value gives us 3x² - 2x - 2. Integrating this expression from -2 to 0 gives us the surface area bounded by the curve and the x-axis.