Finding maximum or minimum of volume of solid revolved about a region

[taekwondo22]

Question:

For c>0, the graphs of y=(c^2)(x^2) and y=c bound a plane region. Revolve this region about the horizontal line y= -(1/c) to form a solid.

For what value of c is the volume of this solid a maximal or minimal (Use calculus 1 techniques).

First, I found the volume of this solid using the washer's method and I got this answer:

∏ [(2c^2 +4)* square root of (1/c) - (2c^4/5)*(1/c)^(5/2) - (4c/3)* (1/c)^ (3/2)]
I know that in order to find the maximum or minimum, I have to find the first derivative of the function and then use the sign chart, etc. But I am not sure whether I will have to use the volume I got above for the solid. Do I just differentiate only y=(c^2)(x^2) and find the critical values?

 

[HallsofIvy]

I don't understand your confusion. You want to find c that will maximize or minimize the volume. Of course, it is the volume you must differentiate, with respect to c, not just one of the curves bounding the region.

 

[BruceW]

I'm not sure whether you got the volume correct, because I can't tell what order the operations should be done (not enough brackets in the answer).

But assuming you have done it right, then you've got the volume which depends on the parameter c. You're trying to find out what the parameter c must be for volume to be maximum.

You're trying to find the max of the volume, so this is a big clue as to what you should differentiate. Equivalently, you could just draw a graph of volume against c and find the max/min points. That is all you are doing when you differentiate to find max/min.