Final answer:
The shell method is used to calculate the volume of a solid by revolving the region between y = 2x + 3 and y = x^2. Intersection points are found at x = 3 and x = -1. Integrate cylindrical shells' volumes using V = 2πx((2x + 3) - x^2) from x = -1 to x = 3.
Step-by-step explanation:
To use the shell method to find the volume of the solid generated by revolving the region bounded by the line y = 2x + 3 and the parabola y = x^2, we first need to find the intersection points of the two curves. This is where they meet and form boundaries of the region to be revolved.
To find the intersection points, set the two equations equal to each other and solve for x:
2x + 3 = x^2
Rearranging gives x^2 - 2x - 3 = 0, which factors to (x - 3)(x + 1) = 0. Therefore, the intersection points are x = 3 and x = -1. Now, applying the shell method involves integrating cylindrical shells' volumes from the lower to upper bounds of x, which in this case are -1 and 3.
The formula for the volume of a shell is V = 2πrh, where 'r' is the radius and 'h' is the height of the shell. In this case, r = x, and h is the difference between the functions, (2x + 3) - x^2. Therefore, the volume of the solid V is:
V = ∫-13 2πx((2x + 3) - x^2) dx
Carrying out this integration will yield the volume of the solid.