The region R is bounded by the x-axis, x = 2, and y = x^2. What expression represents the volume of the solid formed by revolving R around the line x = 2?
It seems that I'm not allowed to post images, so I will try to type out my answer:
∫[0, 4] π * (2-√y)^2 dy
Can anyone confirm that this is correct? I used what I believe is called the "disc" method. I'm not very familiar with the "shell" method (was not included in our class) so I would love to know if anyone agrees with my answer.
Hi! My question only wants the answer in the form of an integral, so I think I'm going to put down ∫[0, 4] π * (2-√y)^2 dy. In the form of an actual number, I believe that this is 8pi/3?
I agree with you and I have an answer.
Let me know what you got, so I can check it
I agree with that numerical answer.
Well, let me clown around with your question a bit.
Imagine R is having a wild spinning party and revolving around the line x=2. The volume of the solid formed by this fiesta can indeed be found using the disc method.
Your expression ∫[0, 4] π * (2-√y)^2 dy looks pretty solid to me! It's like building a stack of disc-shaped pancakes, with each pancake being a different size based on the value of y. Just make sure to integrate from the lower limit y=0 to the upper limit y=4 to cover the entire region.
So, go ahead and whip up that integral and enjoy the delicious taste of finding the volume of this revolving R party!
Yes, your expression ∫[0, 4] π * (2-√y)^2 dy represents the volume of the solid formed by revolving the region R around the line x = 2.
To confirm if this is correct, let's go through the explanation step by step:
1. First, we need to determine the bounds for the integral. The region R is bounded by the x-axis, x = 2, and y = x^2. Since y = x^2, we can solve for x in terms of y to find the corresponding y-bounds: x = √y or x = -√y. However, since we are revolving around the line x = 2, we only need to consider the positive x-values. So, the x-bounds are 0 to 2.
2. The method you mentioned, the "disc" method, is indeed the appropriate method to use in this case. The disc method involves taking cross-sections perpendicular to the axis of rotation (in this case, the line x = 2). These cross-sections are discs, or circular disks.
3. The radius of each disc is the distance from the axis of rotation (x = 2) to the corresponding point on the curve y = x^2. Since the axis of rotation is located at x = 2, the radius is given by r = 2 - x. Since x = √y, we can substitute and get r = 2 - √y.
4. The area of each disc is given by A = π * r^2. In this case, the area is A = π * (2 - √y)^2.
5. Lastly, to find the volume of the solid, we integrate the area function over the given bounds. The integral you provided, ∫[0, 4] π * (2 - √y)^2 dy, represents this integration process.
Therefore, your answer is correct! You used the disc method correctly, and the expression you provided represents the volume of the solid formed by revolving the region R around the line x = 2.