This question was originally posted on another student's thread.
The cup of a wine glass has a shape formed by rotating the parabola y=x^2 about the y-axis. Its upper rim is a circle of radius 1 unit . How much wine can it hold ?
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posted by Reina
today at 3:09pm
clearly, the volume of the cup is found by rotating the curve
y = x^2
about the y-axis. Since the radius at the top is 1, the curve is from (0,0) to (1,1)
Using discs of thickness dy, we have
v = ∫[0,1] πr^2 dy
where r = x
v = ∫[0,1] πy dy = π/2
Makes sense, since a cone of that radius would have volume π/3. and the goblet, being round instead of pointed, would hold more.
picture the cup as a stack of discs
the thickness of a disc is dy ... the radius of a disc is x
... the volume of a disc is ... π x^2 dy
... y = x^2 ... dv = π y dy
integrating from y = 0 to y = 1 ... v = π y^2 / 2
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.