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# Compute the volume of a tetrahedron with given vertices

Consider the tetrahedron with vertices at the origin and at the points where the plane

intersects the coordinate axes. Compute the volume of this tetrahedron.

First, the intercepts of the plane are given by . Then from a previous exercise (Section 13.14, Exercise #13) we know that the volume of a tetrahedron with vertices is

Letting we have

# Prove a vector formula for the volume of a tetrahedron

1. Consider a tetrahedron with vertices . Prove that the volume of the tetrahedron is given by the formula

2. Compute the volume in the case that

1. Proof. We know the volume of a tetrahedron is given by (where denotes the altitude of the tetrahedron). We know (page 490 of Apostol) that the volume of the parallelepiped with base formed by vector and height formed by vector is given by . In this case we have that the base of the tetrahedron is formed by the vectors and , and the height is formed by the vector . Further, we know that the area of the base described by the vectors and is one half that of the parallelepiped whose base is given by vectors and (since the base of the parallelepiped described by vectors and is a rectangle, and the base of the tetrahedron is the triangle formed by cutting this rectangle along the diagonal). Therefore we have

2. Using the formula in part (a) with the given values of we have

# Compute the volume of a parallelpiped determined by given vectors

Compute the volume of the parallelpiped determined by the vectors .

The volume is given by the scalar triple product

# Compute the area and volume of solids of revolution of e-2x

Define the function for all . Let

Compute

1. A(t);
2. V(t);
3. W(t);
4. .

1. The area of the ordinate set on is given by the integral,

2. The volume of the solid of revolution obtained by rotating about the -axis is

3. To compute the volume of the solid of revolution obtained by rotating about the -axis we first find as a function of .

Since , the integral is then from to 1 and we have

4. Finally, using parts (c) and (d) we can compute the limit,

# Find the slope and area under the graph for a given function

Let

1. Determine the slope of the graph of at the point with -coordinate 1.
2. Find the volume of the solid of revolution formed by rotating the region between the graph of and the interval about the -axis.

1. To take this derivative, using logarithmic differentiation will be easier,

Then differentiating both sides we have,

So, to find the slope at the point with we evaluate,

2. First, the integral to compute the volume of the solid of revolution is,

To evaluate this we use the partial fraction decomposition,

This gives us the equation

Evaluating at , , and we obtain

Therefore, we have

# Find a function given the volume of a solid determined by the function

Given a solid with base defined by the ordinate set of a continuous function on the interval . The cross sections take perpendicular to are in the shape of squares. Find the function if the volume of the solid is

The volume of the solid is equal to the integral,

since the cross sections are in the shape of squares and the length of the base is . So the cross sectional area at each point from 1 to is , and then the volume is obtained by integrating these areas over . Setting this equal to the given formula for the volume,

# Compute the volume of the solid of revolution generated by a region bounded by inequalities

Define a region to be the points such that

Sketch this region, and compute the volume of the solid of revolution under the following revolutions:

3. Revolution about the vertical line through the point .
4. Revolution about the horizontal line through the point .

First, the sketch of the region is:

1. We compute the volume from revolving about the -axis as follows,

2. First, we find an equation for in terms of .

Then, we compute the volume as follows from revolving about the -axis as follows,

3. We compute the volume as follows from revolving about the vertical line as follows,

4. We compute the volume as follows from revolving about the horizontal line as follows,

# Establish the prismoid formula

Given a solid whose cross sections have area for . Compute the volume of this solid in terms of , the cross sections corresponding to , respectively.

Since the area of the cross section at is given by , we have

Then, we compute the volume,

# Compute the volume of a solid with given properties

Given a solid with square cross sections perpendicular to the -axis and with their center on the -axis. If the cross section square at has side length , compute the volume of the solid for .

The area of the cross sections is given by

So, the volume of the solid for is given by

# Compute the volume of a solid with given properties

Given a solid with circle base of radius 2 and cross sections which are equilateral triangles, compute the volume of the solid.

We may describe the top half of the circular base of the solid by the equation

Thus, the length of the base of any equilateral triangular cross section is

Since these are equilateral triangles with side length , the area is given by

Then we compute the volume,

(Note: Apostol gives the solution in the back of the book, but I keep getting , as does Edwin in the comments. I’m marking this as an error in the book for now. If you see where my solution is wrong and Apostol is correct please leave a comment and let me know.)