The problem involves finding the mass of a solid enclosed by a paraboloid defined by the equation z = x² + y² and a plane at z = 9, with a density function given by f(x,y,z) = x². The context is within the subject area of multivariable calculus, specifically dealing with triple integrals in cylindrical coordinates.
There is an ongoing examination of the limits of integration, with some participants suggesting corrections to the initial attempts. Guidance has been offered regarding the relationship between r and z, indicating a productive direction in clarifying the setup of the problem.
Participants note that the limits for r should depend on the paraboloid, and there is a discussion about the implications of integrating in cylindrical coordinates. The conversation reflects a need to reconcile the geometric interpretation of the solid with the mathematical formulation.
Stan12 said:Homework Statement
Let E be the solid enclosed by the paraboloid z = x2 + y2 and the plane z = 9. Suppose the density of this solid at any point (x,y,z) is given by f(x,y,z) = x2.
Homework Equations
x2 + y2 = r2 = 9; r = 3
∫∫∫E x2
The Attempt at a Solution
The limit of z is given z=9, found r = 3, and θ=2∏
x = rcosθ
y = rsinθ
z = z
∫∫∫ r2cos2θ r dzdrdθ
I got 729∏ / 4 as final answer
LCKurtz said:I don't think that is correct. Show us your limits and your work.
Stan12 said:z = x2 + y2 and plane z = 9
I set x2 + y2 = r2
and found that 0<r<3
and 0<z<9 since it's given
and 0<θ<2∏
x=rcosθ in cylindrical coord.
f(x,y,z) = x2 --> x2 in cylindrical coord. = (rcosθ)2
I set the function under a triple integral with restriction above
so I got ∫∫∫ (rcos)2 rdrdzdθ
Stan12 said:So the limits to r is from 0 to r = √x2 + y2 or r = √z
and limit of z is from 0 to 9 ? since the paraboloid begins at 0 and enclosed by the plane at z = 9
∫∫∫ r2cos2θ rdrdzdθ