To find the amount of wine that the cup can hold, we need to determine the volume of the cup. Here's how we can approach the problem:
1. Determine the equation of the parabola: The cup is formed by rotating the parabola y = x^2 about the y-axis. Since the upper rim of the cup is a circle with a radius of 1 unit, we can determine the intersection points of the parabola and the circle.
To find these intersection points, we substitute the equation of the circle, x^2 + y^2 = 1, into the equation of the parabola: y = x^2. This gives us the following equation: x^2 + (x^2)^2 = 1. Simplify and solve for x:
x^2 + x^4 = 1
x^4 + x^2 - 1 = 0
We can solve this quadratic equation to find the values of x. By finding the appropriate intersection points, we can determine the bounds of integration for the following step.
2. Find the volume of the cup: To find the volume of the cup, we need to integrate the cross-sectional area function with respect to x. Since the shape of the cross-section is a disk (due to rotating the parabola), the cross-sectional area can be represented by A = πy^2.
Integrate this equation from the lower bound x = a to the upper bound x = b (the x-values of the intersection points) to find the volume V:
V = ∫[a, b] πy^2 dx
3. Evaluate the integral: Substitute y = x^2, and solve the integral to find the volume V:
V = ∫[a, b] π(x^2)^2 dx
4. Calculate the final answer: Once you have evaluated the integral, you will have the volume of the cup. This value represents how much wine the cup can hold.
Keep in mind that this explanation outlines the process to find the answer. You will need to carry out the calculations to get the actual value.