Final answer:
The volume of the solid generated by revolving the region bounded by the curves y = x² and y = 6x - x² around the x-axis is found using disk integration, with the limits of integration determined by the points of intersection of the curves (x = 0 and x = 3).
Step-by-step explanation:
To find the volume of the solid generated by revolving the region bounded by the given graphs around the x-axis, we need to apply the method of disk integration. We have two functions: y = x² and y = 6x - x². To generate the solid, we revolve the area between these curves around the x-axis. To properly apply this method, we need to calculate the volume of a typical disk and then integrate from the x-coordinate where the curves intersect on the left to where they intersect on the right.
First, we set the functions equal to each other to solve for the points of intersection: x² = 6x - x², this leads to 2x² = 6x, and thus x = 0 or x = 3. These are our limits of integration.
Next, we determine the volume of a typical disk with inner radius ri = x² and outer radius ro = 6x - x². The volume of the disk is π(ro² - ri²) ∆x. Finally, we integrate this expression from x = 0 to x = 3 to find the total volume.
The integral looks like this: ∫0³ π((6x - x²)² - (x²)²) dx. Solving this integral will give us the total volume of the solid of revolution.