To calculate the volume of the solid within a sphere and outside a cylinder, we use the volume formulas for both shapes and polar coordinates for integration. The total volume of the sphere is found using the formula V = 4/3 πr³, then we subtract the volume of the cylinder inside the sphere, taking into account the changing height due to the sphere's curvature.
- To find the volume of the given solid, which is inside the sphere x² + y² + z² = 25 and outside the cylinder x² + y² = 1, we will use polar coordinates for the integration.
- Using the formula for the volume of a sphere, V = 4/3 πr³, we can calculate the volume of the sphere with radius 5 (since the equation x² + y² + z² = 25 corresponds to a sphere of radius √25, which is 5).
- The volume inside the sphere but outside the cylinder is the total volume of the sphere minus the volume occupied by the cylinder within the sphere.
- To find the volume of the cylinder, we use the formula V = πr²h.
- However, we must consider the height of the cylinder that lies within the sphere, which will vary along the cylinder's axis because of the sphere's curvature.
- This requires setting up an integral that accounts for the changing height.
- To solve the problem, we would integrate the volume element dV in polar coordinates over the appropriate limits that define the region inside the sphere but outside the cylinder.
- The resulting integral, after evaluating, gives the desired volume.