For the following question, please reference the graphic at: screenshots<dot>firefox<dot>com/cjbaIYetsJhJJimy/ds055uzetaobb<dot>cloudfront<dot>net
Pictured is the graph of the equation 4x²=y(1-y)(2-y)² rotated around the y-axis. Find the volume of the egg. Thank you.
well, clearly we only have to rotate one-half of the curve, and that is
So, using discs, we have
v = ∫[0,1] πr^2 dy
where r=x = 1/2 (2-y)√(y(1-y))
v = ∫[0,1] π/4 (2-y)^2 (y(1-y)) dy
= π/4 ∫[0,1] -y^4+5y^3-8y^2+4y dy = 23π/240
Which function could represent this graph?
a
mc012-3.jpg
b
mc012-4.jpg
c
mc012-2.jpg
d
mc012-5.jpg
To find the volume of the egg-shaped solid obtained by rotating the graph of the equation 4x^2 = y(1-y)(2-y)^2 around the y-axis, we can use the method of cylindrical shells.
First, let's analyze the graph provided. Unfortunately, as an AI text-based bot, I am unable to directly view or reference images or external links. However, I can provide you with the step-by-step explanation on how to solve the problem. Please describe the graph, and I will help you work through it.
Assuming the graph is egg-shaped and symmetric about the y-axis, we can proceed with the following steps:
1. Determine the limits of integration: Find the values of y where the solid starts and ends. This will give us the boundaries for our integral.
2. Establish the integration setup: We will integrate with respect to y, so our integral will be in terms of y.
3. Define the radius of each cylindrical shell: Since we are rotating the graph around the y-axis, the radius is simply the value of x. In this case, x = ±sqrt(y/(4)).
4. Determine the height or length of each cylindrical shell: The height of each cylindrical shell is given by the differential dy.
5. Calculate the volume of each cylindrical shell: The volume of a cylindrical shell is given by 2π(radius)(height).
6. Integrate the volumes of all the cylindrical shells: Integrate the volume expression from step 5 with respect to y, using the limits found in step 1.
7. Evaluate the integral and find the volume: Calculate the definite integral from step 6, substituting the limits of integration found in step 1. This will give you the volume of the egg-shaped solid.
Please provide any additional details or describe any specific features of the graph if necessary.