Final answer:
To find the volume of the solid inside the sphere with radius 4 and outside the cylinder with radius 2, calculate the volumes of both using their respective formulae and subtract the volume of the cylinder from the volume of the sphere.
Step-by-step explanation:
Finding the Volume of a Solid Using Polar Coordinates
To find the volume of a given solid that is inside the sphere x²+y²+z²=16 and outside the cylinder x²+y²=4 using polar coordinates, we first recognize the spherical radius as 4 (since the equation is r² = 16). The radius of the cylinder is 2 because the equation is r² = 4.
Using the formula for the volume of a sphere and recognizing the volume is a function of the cube of the radius, we can calculate the volume of the sphere as V = 4/3 (pi) (r)³ = 4/3 (pi) (4)³. To find the volume of the specified solid, we need to subtract the volume of the cylinder inside the sphere.
The volume of the cylinder is the area of its circular base times the height: A_cylinder = (pi) (radius)² * height. Since the height of the cylinder is the diameter of the sphere, we calculate the volume of the cylinder as V = (pi) (2)² * (2 * 4).
The volume of the desired solid is then found by subtracting the volume of the cylinder from the volume of the sphere. Note that since we are dealing with a three-dimensional object in polar coordinates, we consider the spherical coordinates and, typically, integration when calculating the volume of the solid.