Final answer:
To find the volume of the solid, we can use the shell method. This involves integrating the area of thin cylindrical shells. The formula to find the volume using the shell method is V = 2π∫(x)(f(x)) dx. In this problem, we need to evaluate the integral ∫(x)(2x) dx from x = 0 to x = 3.
Step-by-step explanation:
To find the volume of the solid generated by revolving the region bound by y = 2x, y = 0, and x = 3 about the y-axis, we can use the shell method. This method involves integrating the area of thin cylindrical shells that are parallel to the axis of rotation.
The formula to find the volume using the shell method is V = 2π∫(x)(f(x)) dx, where f(x) is the distance from the curve to the axis of rotation (in this case, the y-axis).
In this problem, the distance from the curve y = 2x to the y-axis is simply x. So, we need to evaluate the integral ∫(x)(2x) dx from x = 0 to x = 3.
Integrating this expression gives us: V = 2π∫(x)(2x) dx = 2π∫2x^2 dx = 4π∫x^2 dx.
Finally, integrating x^2 gives us: V = 4π * (1/3) * x^3. Evaluating the integral from x = 0 to x = 3 gives us the final volume of the solid generated.