Final answer:
To find the volume of the solid generated by revolving the region bounded by the graphs of y = 2x² + 1 and y = 2x + 5 about the x-axis, use the method of cylindrical shells.
Step-by-step explanation:
To find the volume of the solid generated by revolving the region bounded by the graphs of y = 2x² + 1 and y = 2x + 5 about the x-axis, we can use the method of cylindrical shells.
First, we need to find the limits of integration. To do this, set the two equations equal to each other:
2x² + 1 = 2x + 5
2x² - 2x - 4 = 0
Then, solve for x to find the x-values where the two curves intersect.
Next, we can set up the integral:
V = 2π∫[a, b] x*(f(x) - g(x)) dx
where a and b are the x-values where the curves intersect, f(x) is the upper curve (2x + 5), and g(x) is the lower curve (2x² + 1).
Finally, evaluate the integral to find the volume of the solid generated by revolving the region bounded by the two curves about the x-axis.