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# 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.

# Compute the volume of the solid of revolution generated by f(x) = x^2

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) = x^(1/4)

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) = x^(1/2)

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 a right circular cone generated by f(x) = cx

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,