We have now learned that we define the double integral of a continuous function over a rectangle as a limit of a double Riemann sum, and that these ideas parallel the single-variable integral of a function on an interval . Moreover, this double integral has natural interpretations and applications, and can even be considered over non-rectangular regions, . For instance, given a continuous function over a region , the average value of ,, is given by
It is natural to wonder if it is possible to extend these ideas of double Riemann sums and double integrals for functions of two variables to triple Riemann sums and then triple integrals for functions of three variables. We begin investigating in Preview Activity 11.7.1.
Consider a solid piece of granite in the shape of a box , whose density varies from point to point. Let represent the mass density of the piece of granite at point in kilograms per cubic meter (so we are measuring ,, and in meters). Our goal is to find the mass of this solid.
Recall that if the density was constant, we could find the mass by multiplying the density and volume; since the density varies from point to point, we will use the approach we did with two-variable lamina problems, and slice the solid into small pieces on which the density is roughly constant.
Partition the interval into 2 subintervals of equal length, the interval into 3 subintervals of equal length, and the interval into 2 subintervals of equal length. This partitions the box into sub-boxes as shown in Figure 11.7.1.
Let be the endpoints of the subintervals of after partitioning. Draw a picture of Figure 11.7.1 and label these endpoints on your drawing. Do likewise with and What is the length of each subinterval for from 1 to 2? the length of ? of ?
The partitions of the intervals , and partition the box into sub-boxes. How many sub-boxes are there? What is volume of each sub-box?
Let denote the sub-box . Say that we choose a point in the th sub-box for each possible combination of . What is the meaning of ? What physical quantity will approximate?
What final step(s) would it take to determine the exact mass of the piece of granite?
Subsection11.7.1Triple Riemann Sums and Triple Integrals
Through the application of a mass density distribution over a three-dimensional solid, Preview Activity 11.7.1 suggests that the generalization from double Riemann sums of functions of two variables to triple Riemann sums of functions of three variables is natural. In the same way, so is the generalization from double integrals to triple integrals. By simply adding a -coordinate to our earlier work, we can define both a triple Riemann sum and the corresponding triple integral.
Partition the interval into subintervals of equal length . Let ,,, be the endpoints of these subintervals, where . Do likewise with the interval using subintervals of equal length to generate , and with the interval using subintervals of equal length to have .
Let be the sub-box of with opposite vertices and for between and , between and , and between 1 and . The volume of each is .
Let be a point in box for each ,, and . The resulting triple Riemann sum for on is
If represents the mass density of the box , then, as we saw in Preview Activity 11.7.1, the triple Riemann sum approximates the total mass of the box . In order to find the exact mass of the box, we need to let the number of sub-boxes increase without bound (in other words, let ,, and go to infinity); in this case, the finite sum of the mass approximations becomes the actual mass of the solid . More generally, we have the following definition of the triple integral.
is the mass of . Even more importantly, for any continuous function over the solid , we can use a triple integral to determine the average value of over ,. We note this generalization of our work with functions of two variables along with several others in the following important boxed information. Note that each of these quantities may actually be considered over a general domain in , not simply a box, .
In the Cartesian coordinate system, the volume element is , and, as a consequence, a triple integral of a function over a box in Cartesian coordinates can be evaluated as an iterated integral of the form
If we want to evaluate a triple integral as an iterated integral over a solid that is not a box, then we need to describe the solid in terms of variable limits.
Set up and evaluate the triple integral of over the box .
Let be the solid cone bounded by and . A picture of is shown at right in Figure 11.7.4. Our goal in what follows is to set up an iterated integral of the form
(11.7.1)
to represent the mass of in the setting where tells us the density of at the point . Our particular task is to find the limits on each of the three integrals.
Figure11.7.4.Left: The cone. Right: Its projection.
If we think about slicing up the solid, we can consider slicing the domain of the solidβs projection onto the -plane (just as we would slice a two-dimensional region in ), and then slice in the -direction as well. The projection of the solid onto the -plane is shown at left in Figure 11.7.4. If we decide to first slice the domain of the solidβs projection perpendicular to the -axis, over what range of constant -values would we have to slice?
If we continue with slicing the domain, what are the limits on on a typical slice? How do these depend on ? What, therefore, are the limits on the middle integral?
Finally, now that we have thought about slicing up the two-dimensional domain that is the projection of the cone, what are the limits on in the innermost integral? Note that over any point in the plane, a vertical slice in the direction will involve a range of values from the cone itself to its flat top. In particular, observe that at least one of these limits is not constant but depends on and .
In conclusion, write an iterated integral of the form (11.7.1) that represents the mass of the cone .
Note well: When setting up iterated integrals, the limits on a given variable can be only in terms of the remaining variables. In addition, there are multiple different ways we can choose to set up such an integral. For example, two possibilities for iterated integrals that represent a triple integral over a solid are
where ,,,,,,, and are functions of the indicated variables. There are four other options beyond the two stated here, since the variables ,, and can (theoretically) be arranged in any order. Of course, in many circumstances, an insightful choice of variable order will make it easier to set up an iterated integral, just as was the case when we worked with double integrals.
Find the mass of the tetrahedron in the first octant bounded by the coordinate planes and the plane if the density at point is given by . A picture of the solid tetrahedron is shown at left in Figure 11.7.6.
where is the solid tetrahedron described above. In this example, we choose to integrate with respect to first for the innermost integral. The top of the tetrahedron is given by the equation
To find the bounds on and we project the tetrahedron onto the -plane; this corresponds to setting in the equation . The resulting relation between and is
If we choose to integrate with respect to for the middle integral in the iterated integral, then the lower limit on is the -axis and the upper limit is the hypotenuse of the triangle. Note that the hypotenuse joins the points and and so has equation . Thus, the bounds on are . Finally, the values run from 0 to 6, so the iterated integral that gives the mass of the tetrahedron is
Setting up limits on iterated integrals can require considerable geometric intuition. It is important to not only create carefully labeled figures, but also to think about how we wish to slice the solid. Further, note that when we say βwe will integrate first with respect to ,β by βfirstβ we are referring to the innermost integral in the iterated integral. The next activity explores several different ways we might set up the integral in the preceding example.
How many different iterated integrals could be set up that are equal to the integral in Equation (11.7.2)?
Set up an iterated integral, integrating first with respect to , then , then that is equivalent to the integral in Equation (11.7.2). Before you write down the integral, think about Figure 11.7.6, and draw an appropriate two-dimensional image of an important projection.
Set up an iterated integral, integrating first with respect to , then , then that is equivalent to the integral in Equation (11.7.2). As in (b), think carefully about the geometry first.
Set up an iterated integral, integrating first with respect to , then , then that is equivalent to the integral in Equation (11.7.2).
Now that we have begun to understand how to set up iterated triple integrals, we can apply them to determine important quantities, such as those found in the next activity.
First, set up an iterated double integral to find the volume of the solid as a double integral of a solid under a surface. Then set up an iterated triple integral that gives the volume of the solid . You do not need to evaluate either integral. Compare the two approaches.
Set up (but do not evaluate) iterated integral expressions that will tell us the center of mass of , if the density at point is .
Set up (but do not evaluate) an iterated integral to find the average density on using the density function from part (b).
Use technology appropriately to evaluate the iterated integrals you determined in (a), (b), and (c); does the location you determined for the center of mass make sense?
The moment of inertia of a solid body about an axis in 3-space relates the angular acceleration about this axis to torque (force twisting the body). The moments of inertia about the coordinate axes of a body of constant density and mass occupying a region of volume are defined to be
The motion of a solid object can be analyzed by thinking of the mass as concentrated at a single point, the center of mass. If the object has density at the point and occupies a region , then the coordinates of the center of mass are given by
Assume ,, are in cm. Let be a solid cone with both height and radius 1 and contained between the surfaces and . If has constant mass density of 5 g/cm, find the -coordinate of βs center of mass.
Set up a triple integral to find the mass of the solid tetrahedron bounded by the xy-plane, the yz-plane, the xz-plane, and the plane , if the density function is given by . Write an iterated integral in the form below to find the mass of the solid.
Set up (but do not evaluate) an iterated integral that represents the mass of . Integrate first with respect to , then , then . A picture of the projection of onto the -plane is shown at left in Figure 11.7.9.
Set up (but do not evaluate) an iterated integral that represents the mass of . In this case, integrate first with respect to , then , then . A picture of the projection of onto the -plane is shown at center in Figure 11.7.9.
Set up (but do not evaluate) an iterated integral that represents the mass of . For this integral, integrate first with respect to , then , then . A picture of the projection of onto the -plane is shown at right in Figure 11.7.9.
Which of these three orders of integration is the most natural to you? Why?
Set up, but do not evaluate, an iterated integral expression whose value is the average sum of all real numbers ,, and that have the following property: is between 0 and 2, is greater than or equal to 0 but cannot exceed , and is greater than or equal to 0 but cannot exceed .
Set up, but do not evaluate, an integral expression whose value represents the average value of over the solid region in the first octant bounded by the surface and the coordinate planes ,,.
How are the quantities in (a) and (b) similar? How are they different?
By eliminating the variable , determine the curve of intersection between the two paraboloids, and sketch this curve in the -plane.
Set up, but do not evaluate, an iterated integral expression whose value determines the mass of the solid, integrating first with respect to , then , then . Assume the the solidβs density is given by .
Set up, but do not evaluate, iterated integral expressions whose values determine the mass of the solid using all possible remaining orders of integration. Use as the density of the solid.
Set up, but do not evaluate, iterated integral expressions whose values determine the center of mass of the solid. Again, assume the the solidβs density is given by .
Which coordinates of the center of mass can you determine without evaluating any integral expression? Why?
Consider the solid created by the region enclosed by the circular paraboloid over the region in the -plane enclosed by and the circle in the first, second, and fourth quadrants. Determine the solidβs volume.
Consider the solid region that lies beneath the circular paraboloid over the triangular region between ,, and . Assuming that the solid has its density at point given by , measured in grams per cubic cm, determine the center of mass of the solid.
In a certain room in a house, the walls can be thought of as being formed by the lines ,,, and , where length is measured in feet. In addition, the ceiling of the room is vaulted and is determined by the plane . A heater is stationed in the corner of the room at and causes the temperature in the room at a particular time to be given by
What is the average temperature in the room?
Consider the solid enclosed by the cylinder and the planes and . Assuming that the solidβs density is given by , find the mass and center of mass of the solid.