jsun2015 said:Homework Statement
Find the volume of the solid generated by revolving the region bounded by the parabola y=x^2 and the line y=1 about the line y=1
Homework Equations
V= integral of pi*r^2 from a to b with respect to variable "x"
The Attempt at a Solution
pi(integral of 1-(x^2-1)^2 from 0 to 1 dx)
but The answer is 15pi/16
Dick said:Why do you think the 'r^2' part in your volume equation is 1-(x^2-1)^2 and why do you think the limits to the integration are 0 to 1?
jsun2015 said:r^2
1 represents outer radius (the larger radius)
x^2-1 represents the inner radius (smaller radius)
limits to integration
the radius of the sphere is on the region x= 0 to 1
Dick said:That's confusing me. The formula you gave was actually for the method of disks, which is what I would use here. If you are rotating the region between y=x^2 and y=1 around y=1, then the inner radius is 0, isn't it? And I don't see why you are putting one of the limits to 0. y=x^2 and y=1 cross at x=1 and x=(-1), yes?
jsun2015 said:Yes. Yes. I came up with my original answer because at the outer region 1= radius so we have one and at the inner region 0 = radius as (1^2-1)^2=0
I have only studied the washer method.
Dick said:The disk method is the same as the washer method with an inner radius of 0. Isn't the outer radius ALWAYS 1-x^2? And the inner radius 0?
The volume of a solid is the amount of space that the solid occupies.
The volume of a solid can be calculated by multiplying the length, width, and height of the solid.
Yes, the volume of a solid can be measured in different units such as cubic centimeters, cubic inches, or liters.
The formula for finding the volume of a cube or rectangular solid is V = l x w x h, where l is the length, w is the width, and h is the height.
To find the volume of an irregularly shaped solid, you can use the displacement method. This involves submerging the solid in water and measuring the amount of water displaced, which is equal to the volume of the solid.