Use polar coordinates to find the volume of the given solid.
Inside the sphere x^2+y^2+z^2=25 and outside the cylinder x^2+y^2=1
To find the volume of the solid, we need to determine the region of integration in polar coordinates and then set up the triple integral.
First, let's analyze the equations:
1. The sphere equation, x^2 + y^2 + z^2 = 25, represents a solid sphere with a radius of 5 units centered at the origin (0,0,0).
2. The cylinder equation, x^2 + y^2 = 1, represents an infinitely long cylinder with a radius of 1 unit centered at the origin.
To find the region of integration, we need to determine the limits for the polar coordinates (r, θ).
In polar coordinates, the sphere equation becomes: r^2 + z^2 = 25.
To convert the cylinder equation to polar coordinates:
x = r * cos(θ)
y = r * sin(θ)
Substituting these into the cylinder equation:
(r * cos(θ))^2 + (r * sin(θ))^2 = 1
r^2 * (cos^2(θ) + sin^2(θ)) = 1
r^2 = 1
Therefore, the limits for r are 0 to 1.
To find the limits for θ, we need to consider the entire circumference of the cylinder within the sphere. Since the cylinder is infinitely long, the limits for θ are from 0 to 2π (a full revolution).
Now that we have determined the region of integration in polar coordinates, we can set up the triple integral:
V = ∫∫∫ dV
The limits for the triple integral are:
r: 0 to 1
θ: 0 to 2π
z: -√(25 - r^2) to √(25 - r^2)
The integrand, dV, represents the differential volume element, which is equal to r * dz * dr * dθ.
Therefore, the integral to find the volume becomes:
V = ∫∫∫ r * dz * dr * dθ
Evaluating this triple integral will give us the volume of the given solid.