Final answer:
The volume of the solid obtained by rotating the region bounded by y = 7 - x² and y = 3 about the x-axis is calculated using an integral with washer method, integrating the difference of squares of the functions from the limits obtained by intersection points.
Step-by-step explanation:
To find the volume V of the solid obtained by rotating the region bounded by the curves y = 7 - x² and y = 3 about the x-axis, we use the method of discs or washers.
The region between these two curves forms the shape we will rotate. First, we need to find the points of intersection between the curves, setting 7 - x² = 3 and solving for x, giving us x = ±2. These are the limits of integration for our volume integral.
Next, we consider a vertical slice of the region at a general position x. If we rotate this slice about the x-axis, it creates a washer with an inner radius r1 = 3 and an outer radius r2 = 7 - x². The area A of this washer is π(r2² - r1²).
The volume of the solid is found by integrating this area from -2 to 2:
V = ∫_{-2}^{2} π((7 - x²)² - 3²) dx
This integral calculates the total volume of the solid after the rotation. It can be solved either by hand or with the aid of a calculator or computer algebra system.