Did you draw a sketch? Actually, two sketches would be useful - one of the region defined by the two curves, and another that shows the solid of revolution.Jbreezy said:Homework Statement
Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis.
Homework Equations
y = x^2 +1 , y = 9-x^2: about y = -1
The Attempt at a Solution
I used disks.
Jbreezy said:I said that r = 1 + ((9-x^2)-(x^2 +1 )) = 9 - 2x^2
So my integral all together is
V = ∏ ∫ (9 - 2x^2)^2 dx from [-2,2]
Does this set up look correct.?
Looks OK to me. It would have been helpful to indicate that your using shells.Jbreezy said:Also I have another one I would like to confirm.
This one your supposed to set up the integral not evaluate for the volume of a solid.
y = tanx, y = x, x = pi/3 : about the y axis.
I have
V = 2∏∫ x(tanx - x) dx between 0 and ∏/3
I think it is good but just to be safe.
Mark44 said:Did you draw a sketch? Actually, two sketches would be useful - one of the region defined by the two curves, and another that shows the solid of revolution.
Disks won't work, but washers will. A washer is a disk with a hole in its middle. For a typical volume element, the outer radius goes from the line y = -1 to the curve y = 9 - x2. The inner radius goes from the line y = -1 to the curve y = x2 + 1.
Your radii are correct but the integral isn't. The volume of a washer is ##\pi(R^2 - r^2)(\text{thickness})##.Jbreezy said:Yeah I always draw it. Sorry for the confusion with the other problem not indicating I was using shells.
So what about this. r = inner radius, R = outer radius
r = 1 + ( x^2 + 1) R = 1+ (9-x^2)
So V = ∏∫( x^2 +2)^2 + (10-x^2)^2 dx I think that is it but I have a little trouble with the radius sometimes.
There are a few ways to determine if you have set up your experiment correctly. First, make sure you have followed the instructions or protocols provided by your experiment's design. Second, double-check that you have all the necessary materials and equipment. Finally, if possible, have a colleague or mentor review your setup to ensure accuracy.
Some common mistakes when setting up an experiment include not following the instructions or protocols, not properly calibrating equipment, and not controlling for external variables. It is also important to accurately record all steps and data during the setup process to prevent errors.
If you encounter issues with your experiment setup, first check for any obvious errors or mistakes. Then, refer to the instructions or protocols to ensure you are following the correct procedures. If the issue persists, consult with a mentor or colleague for further troubleshooting steps.
If you realize that you have set up your experiment incorrectly, do not panic. Stop the experiment and consult with a mentor or colleague for guidance. Depending on the severity of the mistake, you may be able to make adjustments to the setup or may need to start over.
Yes, there are several safety precautions to keep in mind when setting up an experiment. Always wear appropriate personal protective equipment, such as gloves or goggles, when handling chemicals or hazardous materials. Follow proper procedures for disposal of materials and always clean up any spills or messes immediately. If in doubt, consult with a safety officer or mentor for guidance.