Volume -- Multiple and Iterated Integrals

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In this module we develop methods for finding the volume of a solid of the form

Example

Consider the tetrahedron shown in the figure below

The four vertices of this tetrahedron are located at the points:

Thus, its base can be described by

Approximating D by many small squares

The first method is based on the movie at the left. We begin by breaking the region D up into many small squares and then above each square we erect a prism whose height is given by the function f(x, y) at a point in the square. The volume of each of these prisms is the area of its base multiplied by its height. By adding the volumes of these prisms we obtain an estimate for the volume of the tetrahedron. By using many very small squares we can obtain a very good estimate of the volume of the tetrahedron. The exact volume of the tetrahedron is given by the limit of these estimates.

Your CAS window contains this example. Look at it to make sure you understand the details and then modify it to estimate the area of the tetrahedron whose four vertices are at the points:

The approach illustrated above is called the Multiple Integral. We begin by making a series of estimates. For any positive integer n we proceed as follows.

• First we divide the xy-plane into squares that are 1/n on a side. We mark the x-axis off at the points
  x_i = i / n

  and the y-axis off at the points

  y_j = j / n

• Next we look at squares whose x-coordinate goes from x_{i - 1} to x_i
  and whose y-coordinate goes from y_{j - 1} to y_j.

• In each of these squares we focus on the lower left corner -- namely,
  (x_{i - 1}, y_{j - 1}).

• We only use those squares for which this point is in the region D.

• For each of these squares we compute the volume of the prism based at this square --
  f(x_{i - 1}, y_{j - 1}) (1 / n)²

• Finally we estimate the volume of our solid by

  

The exact volume is the limit of these estimates. This limit is called the multiple integral of the function f(x, y) over the region D.

On the right side of the formula above each term has two factors.

	measures height in units of length
	measures area in units of length2

Similarly, you should think of the integrand on the left side of this equation as having two factors.

	measures height in units of length
	measures area in units of length2

Slicing the tetrahedron perpendicular to the x-axis

This method is based on the movie on the left. The basic idea is simple -- we slice the solid at various points along the x-axis using slices that are parallel to the yz-plane. If all of these slices had the same area then the volume of the solid would be this area multiplied by the length of the solid measured in the x-direction. Because the area of the slices varies, simple multiplication is replaced by integration. For each value x along the x-axis, the section can be described by z = 2 - x - y/2 with y <= 0 <= 4 - 2 x

Thus, the area of the slice is

Now to determine the volume of the solid we "multiply" the area obtained above by the length of the solid measured along the x-axis -- that is, we integrate,

In general, we can determine the volume of a solid described by

• Identify the units associated with each element of the integral formula above.

• Your CAS window has the example above worked out. Make sure you understand this example and then use your CAS window to find the volume of the tetrehedron whose four vertices are at the points:
  (0, 0, 0), (10, 0, 0), (0, 5, 0), and (0, 0, 8).

This approach is called an iterated integral and we say that the outer integration is with respect to x and the inner integration is with respect to y.

Slicing the tetrahedron perpendicular to the y-axis

This method is based on the movie at the left. The basic idea is simple -- we slice the solid at various points along the y-axis using slices that are parallel to the xz-plane. If all of these slices had the same area then the volume of the solid would be this area multiplied by the length of the solid measured in the y-direction. Because the area of the slices varies, simple multiplication is replaced by integration. For each value y along the y-axis, the section can be described by z = 2 - x - y/2 with x <= 0 <= 2 - y / 2

Thus, the area of the slice is

Now to determine the volume of the solid we "multiply" the area obtained above by the length of the solid measured along the y-axis -- that is, we integrate,

In general, we can determine the volume of a solid described by

Your CAS window has the example above worked out. Make sure you understand this example and then use your CAS window to find the volume of the tetrehedron whose four vertices are at the points:

Note that this is the same solid that you looked at earlier. Your two answers should agree.

This approach is called an iterated integral and we say that the outer integration is with respect to y and the inner integration is with respect to x.

We have solved our basic problem three different ways -- by multiple integration, and by two varieties of iterated integration. In practice, the iterated integration is used most frequently.