Final answer:
The volume of a rugby ball shaped as an ellipsoid obtained by rotating the given ellipse x²/30 + y²/16 = 1 about the x-axis is calculated using the disc method, resulting in a volume of 2560π/3 cubic units.
Step-by-step explanation:
The student asked to find the volume of a rugby ball, which is an ellipsoid created by rotating an ellipse around the x-axis. The given equation of the ellipse is x²/30 + y²/16 = 1. To find the volume of the ellipsoid (rugby ball), we use the formula for the volume of a solid of revolution, specifically the disc method.
The volume V of a solid of revolution obtained by rotating an area around the x-axis is given by:
V = π ∗ ∫_a^b [f(x)]² dx, where f(x) is the function that defines the top half of the area and [a, b] is the interval over which it is being revolved.
In our case, the function representing the top half of the ellipse is f(x) = √(16 - 16x²/30), and it is revolved around the interval [-√30, √30].
We calculate the volume:
- Convert the equation of the ellipse to the form y = f(x).
- Compute the integral using the volume formula.
By solving the integral, we find the volume of the rugby ball to be:
V = π ∗ ∫_{-√30}^{√30} (16 - 16x²/30) dx = π ∗ [16x - (16x³)/90]_{-√30}^{√30}
Upon evaluating the integral, we get the volume of the rugby ball is 4π∗30∗16/3, which simplifies to 2560π/3 cubic units.