Final answer:
To find the volume of the solid obtained by rotating the region bounded by the curves y = 536 - x², y = 0, x = 2, x = 5 about the x-axis, we can use the method of cylindrical shells.
Step-by-step explanation:
To find the volume of the solid obtained by rotating the region bounded by the curves y = 536 - x², y = 0, x = 2, x = 5 about the x-axis, we can use the method of cylindrical shells.
First, we need to determine the height of each cylindrical shell. This is given by the difference between the upper curve y = 536 - x² and the lower curve y = 0. So the height h is h = (536 - x²) - 0 = 536 - x².
The radius of each cylindrical shell is the distance from the x-axis to the curve at each value of x. Since the axis of rotation is the x-axis, the radius is simply x.
The volume of each cylindrical shell is given by dV = 2πxh dx. To find the total volume, we integrate this expression from x = 2 to x = 5:
V = ∫[2,5] 2πx(536 - x²) dx
After integrating and evaluating the integral, we can find the volume of the solid.