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# Compute the volume of a sphere with a cylindrical hole of length 2h removed

Given a solid sphere, drill a cylindrical hole of length through the center of the sphere. Prove that the volume of the resulting ring is , where .

Proof. Let denote the radius of the sphere, and denote the radius of the cylindrical hole. Then the volume of the ring is the volume of the solid of revolution formed by rotating the area between the functions about the -axis. Since the length of the hole is , we know ; thus, . So, we have, # Volume of cylindrical hole removed from a sphere

Given a solid sphere of radius , what is the volume of material from a hole of radius through the center of the sphere.

First, the volume of a sphere of radius is given by Then, the volume of a sphere with a hole drilled in it is the volume of the solid of revolution generated by the region between and from to . Denoting this volume by we then have, Thus, the volume of the material removed from the sphere by drilling a hole in it is given by # Find an interval on which the solid of revolution has a particular volume

Let Find a real number such that the solid of revolution generated by the region between and on the interval has volume .

The volume of the solid of revolution generated by the region between the graphs of and on the interval is, So, we set this equal to and solve for , # Compute the volume of the solid of revolution generated by the region between two functions

Let Sketch the graph of region between these functions and compute the volume of the solid of revolution generated by revolving this region about the -axis.

The sketch of the region between and is as follows: We then compute the volume of the solid of revolution as follows. # Compute the volume of the solid of revolution generated by the region between two functions

Let Sketch the graph of region between these functions and compute the volume of the solid of revolution generated by revolving this region about the -axis.

The sketch of the region between and is as follows: We then compute the volume of the solid of revolution as follows. # Compute the volume of the solid of revolution generated by the region between two functions

Let Sketch the graph of region between these functions and compute the volume of the solid of revolution generated by revolving this region about the -axis.

The sketch of the region between and is as follows: We then compute the volume of the solid of revolution as follows. # Compute the volume of the solid of revolution generated by the region between two functions

Let Sketch the graph of region between these functions and compute the volume of the solid of revolution generated by revolving this region about the -axis.

The sketch of the region between and is as follows: We then compute the volume of the solid of revolution as follows. # Compute the volume of the solid of revolution generated by f(x) = sin x + cos x

Sketch the graph and compute the volume of the solid of revolution generated by: The sketch of the ordinate set of on is as follows: We then compute the volume of the solid of revolution. We reference the previous exercises here and here for the integrals of and . # Compute the volume of the solid of revolution generated by f(x) = cos x

Sketch the graph and compute the volume of the solid of revolution generated by: The sketch of the ordinate set of on is as follows: We then compute the volume of the solid of revolution. # Compute the volume of the solid of revolution generated by f(x) = sin x

Sketch the graph and compute the volume of the solid of revolution generated by: The sketch of the ordinate set of on is as follows: We then compute the volume of the solid of revolution. 