physics&math said:Homework Statement
Find the volume using cylindrical coordinates bounded by:
x2+y2+z2=2 and
z = x2+y2
Homework Equations
Converting to cylindrical coordinates:
z = √2-r2 and
z = r2
The Attempt at a Solution
I figured z would go from r2 to √2-r2
r from 0 to √2
and θ from 0 to 2∏.
However, when I evaluate the integral I get a negative volume. (And no, I did not forget to multiply by an extra r)
I have no idea where I've gone wrong.
The formula for calculating volume in cylindrical coordinates is V = πr2h, where r is the radius and h is the height of the cylinder.
The volume in cylindrical coordinates is equivalent to the volume in Cartesian coordinates, as both systems use the same units of measurement. However, the formula for calculating volume in cylindrical coordinates takes into account the shape of the cylinder, while the formula for calculating volume in Cartesian coordinates does not.
No, volume cannot be negative in any coordinate system. Volume is a measure of the space occupied by an object and cannot have a negative value.
If both the radius and height are doubled, the volume in cylindrical coordinates will increase by a factor of 8. This is because the formula for volume in cylindrical coordinates is V = πr2h, so doubling both r and h results in a volume of 22 * 2 = 8 times the original volume.
No, volume in cylindrical coordinates can only be used to calculate the volume of objects that have a cylindrical shape, such as cylinders, cones, and spheres. For other shapes, different coordinate systems and formulas are needed to calculate volume.