The region W is the cone shown below.
The angle at the vertex is 2π/3, and the top is flat and at a height of 5/sqrt(3). Write the limits of integration for ∫WdV in the following coordinates (do not reduce the domain of integration by taking advantage of symmetry):
(a) Cartesian:
(b) Cylindrical:
(c) Spherical:
(a) Cartesian: To find the limits of integration in Cartesian coordinates, we need to understand the shape of the cone in this coordinate system. The cone is basically a straight circular cylinder with a flat top. Let's first find the limits for the x-coordinate. Since the top of the cone is flat, the x-coordinate limits will be constant throughout the cone. We can simply take the limits as the radius of the circular base of the cone. Let's call that radius A. The limits for x would then be -A to A.
For the y-coordinate, as we move up the cone, the value of y decreases linearly with the height. The height of the cone is given as 5/sqrt(3), so the maximum value of y at any given height can be found using the equation of a straight line. We can express the equation of the line as y = (5/sqrt(3)) - (10/sqrt(3))*(x/A), where x ranges from -A to A. So the limits for the y-coordinate would be ((5/sqrt(3)) - (10/sqrt(3))*(x/A)) to -((5/sqrt(3)) - (10/sqrt(3))*(x/A)).
Finally, for the z-coordinate, the limits will be constant throughout the cone, i.e., 0 to (5/sqrt(3)).
So the limits of integration in Cartesian coordinates would be:
x: -A to A
y: ((5/sqrt(3)) - (10/sqrt(3))*(x/A)) to -((5/sqrt(3)) - (10/sqrt(3))*(x/A))
z: 0 to (5/sqrt(3))
(b) Cylindrical: In cylindrical coordinates, the limits of integration for z will remain the same as in Cartesian coordinates, i.e., 0 to (5/sqrt(3)). The limits for the radius (ρ) will be 0 to A, as the cone is contained within the circular base of radius A.
The angular limits (θ) in cylindrical coordinates will be the same as in Cartesian coordinates, i.e., 0 to 2π.
So the limits of integration in cylindrical coordinates would be:
ρ: 0 to A
θ: 0 to 2π
z: 0 to (5/sqrt(3))
(c) Spherical: In spherical coordinates, the radial distance (r) ranges from 0 to the height of the cone, which is (5/sqrt(3)).
The polar angle (θ) ranges from 0 to 2π, covering the whole cone.
The azimuthal angle (φ) depends on the value of r. For this cone, the φ angle is irrelevant since the top is flat and there is no rotation required around the z-axis.
So the limits of integration in spherical coordinates would be:
r: 0 to (5/sqrt(3))
θ: 0 to 2π
φ: Not applicable, as φ does not affect the shape of the cone
To find the limits of integration for ∫WdV in different coordinate systems, we can use the given information about the cone.
(a) In Cartesian coordinates, the limits of integration depend on the shape of the region W. Since the top of the cone is flat, we can use a bounded rectangular region in the xy-plane.
Let's consider the cone apex as the origin (0,0,0). The height of the flat top is given as 5/sqrt(3), which means the plane that bounds the top can be described by the equation z = 5/sqrt(3).
To find the limits of integration in the xy-plane, we need to consider the projection of the cone base onto this plane. The angle at the vertex is 2π/3, so the distance from the origin to the cone base is (5/sqrt(3)) / tan(π/3) = 10/sqrt(3).
Therefore, the limits of integration in Cartesian coordinates are:
x: -10/sqrt(3) ≤ x ≤ 10/sqrt(3)
y: -10/sqrt(3) ≤ y ≤ 10/sqrt(3)
z: 0 ≤ z ≤ 5/sqrt(3)
(b) In cylindrical coordinates, we can use the same limits of integration for the radial coordinate and the height coordinate since the cone is symmetric about the z-axis.
The radial coordinate (ρ) represents the distance from the z-axis, and the height coordinate (z) represents the vertical position.
Given the limits of integration in Cartesian coordinates, we can convert them to cylindrical coordinates:
ρ: 0 ≤ ρ ≤ 10/sqrt(3)
θ: 0 ≤ θ ≤ 2π
z: 0 ≤ z ≤ 5/sqrt(3)
(c) In spherical coordinates, the limits of integration depend on the shape of the region W.
We can still use the same limits for the radial coordinate (ρ) and the azimuthal angle (φ) as in cylindrical coordinates.
The polar angle (θ) represents the angle from the positive z-axis. Since the cone is symmetric about the z-axis, the polar angle ranges from 0 to π/3.
Therefore, the limits of integration in spherical coordinates are:
ρ: 0 ≤ ρ ≤ 10/sqrt(3)
φ: 0 ≤ φ ≤ 2π
θ: 0 ≤ θ ≤ π/3
(a) To find the limits of integration in Cartesian coordinates, we need to determine the range of x, y, and z values for the region W.
Since the top of the cone is flat and at a height of 5/sqrt(3), the z-coordinate ranges from 0 to 5/sqrt(3).
The base of the cone forms a circle centered at the origin with a radius r. To find the value of r, we can use the fact that the angle at the vertex is 2π/3.
For a cone with a circular base, the radius r is related to the height h and the angle θ at the vertex as follows: r = (h/tan(θ/2)).
In this case, h is 5/sqrt(3) and θ is 2π/3. Plugging these values into the formula, we get r = (5/sqrt(3))/tan((2π/3)/2) = (5/sqrt(3))/tan(π/3) = (5/sqrt(3))/(sqrt(3)/3) = 5.
Therefore, the bounds for x and y are -5 to 5. Thus, the limits of integration in Cartesian coordinates are:
x: -5 to 5
y: -5 to 5
z: 0 to 5/sqrt(3)
(b) To determine the limits of integration in cylindrical coordinates, we consider that the base of the cone forms a circle centered at the origin with a radius of 5. Therefore, the limits of integration for the radius coordinate, rho, are 0 to 5.
The angle coordinate, theta, ranges from 0 to 2π since the cone extends for a full revolution around the z-axis.
The z-coordinate ranges from 0 to 5/sqrt(3), as mentioned before.
Therefore, the limits of integration in cylindrical coordinates are:
rho: 0 to 5
theta: 0 to 2π
z: 0 to 5/sqrt(3)
(c) In spherical coordinates, we need to determine the limits of integration for the radial distance, rho, the polar angle, theta, and the azimuthal angle, phi.
The radial distance, rho, ranges from 0 to 5, as the cone extends from the origin to the flat top.
The polar angle, theta, spans from 0 to 2π, representing a full revolution around the z-axis.
The azimuthal angle, phi, depends on the cone's shape and orientation. Since the top of the cone is flat, phi ranges from 0 to the vertical angle at the cone's apex, which is 2π/3.
Therefore, the limits of integration in spherical coordinates are:
rho: 0 to 5
theta: 0 to 2π
phi: 0 to 2π/3
The cone has a top radius of 5.
In Cartesian coordinates, z^2 = (x^2+y^2)/6
∫W dV = ∫[-5,5]∫[-√(25-x^2),√(25-x^2) ∫[0,√((x^2+y^2)/6)] dz dy dz
In cylindrical coordinates,
∫[0,2π] ∫[0,5] ∫[0,r/√3] r dz dr dØ
In spherical coordinates, don't recall the transformations right off, but I'm sure your text has examples. I'll have to think on it a bit.