I have a solid that is comprised of the "solid of revolution" of two functions. I want to find the centroid of the entire solid. Because it is a solid of revolution, I can assume that the x and z coordinates of the centroid will be 0, so what I'm looking for is just the y coordinate of the centroid.
The two functions, if this matters, can be described as follows (shown below):
f(x): a line from (0, 4) to (3, 0). When revolved around the y-axis, this is just a cone
g(x): a parabola (but note that it is flipped over the x-axis). When revolved around the y-axis, it looks something like a rounded bottle cap.
To find the volume of each, I've used the disc-based method of finding the area of revolution.
To find the y-coordinate of the centroid, I have this formula:
$$ \bar y = (1/A) * \int^b_a ((1/2)*f(x)^2) dx $$
This is for a 2D plane, so I'm assuming I can use-- and need to use-- volume instead of area (A). Please correct me if I'm wrong.
(Going forward, note that the density is constant throughout the entire shape.)
Using that formula, I can find the y-coordinate centroid for each of the two solids of revolution. But once I get to that point, how do I get the centroid of the entire shape? Do I just add their centroids and divide by 2? Or is there a complication since g(x) is under the x-axis instead of above it?
Sorry if I provided way more information than is necessary. My secondary goal in asking this is to make sure I'm not making any major conceptual mistakes by using a formula incorrectly or something. :)