Final answer:
To find the volume of the solid obtained by rotating the region bounded by the curves y = 9 - 9x² and y = 0 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, we need to use the method of cylindrical shells.
The equation of the curve y = 9 - 9x² represents a parabola that opens downward. The curve intersects the x-axis at x = -1 and x = 1.
When we rotate the region bounded by these curves about the x-axis, we can think of it as stacking up an infinite number of thin cylinders with height dx. The radius of each cylinder is given by the equation r = y, where y is the distance from the curve to the x-axis.
The volume of each cylindrical shell is given by dV = 2πrhdx, where r is the radius and h is the height. Integrating this from x = -1 to x = 1 will give us the total volume of the solid.
V = ∫(from -1 to 1) (2π(9 - 9x²)x)dx