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.
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.
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.
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.
Consider the ordinate set of the function on the interval
. Revolving this ordinate set about the
-axis generates a right circular cone. Using integration, compute the volume of this right circular cone. Show that the result agrees with our usual formula for the volume of a right circular cone, namely,
, where
is the area of the base.
Proof. First, the area of a cross-section of this solid of revolution is . So, using the formula for the volume of a Cavalieri solid (Theorem 2.7 in Apostol), we have
Since the area of the base is (since
is the height of
at
, and this is then rotated around the
-axis), we have,