In single-variable calculus, recall that we approximated the area under the graph of a positive function on an interval by adding areas of rectangles whose heights are determined by the curve. The general process involved subdividing the interval into smaller subintervals, constructing rectangles on each of these smaller intervals to approximate the region under the curve on that subinterval, then summing the areas of these rectangles to approximate the area under the curve. We will extend this process in this section to its three-dimensional analogs, double Riemann sums and double integrals over rectangles.
Review the concept of the Riemann sum from single-variable calculus. Then, explain how we define the definite integral of a continuous function of a single variable on an interval . Include a sketch of a continuous function on an interval with appropriate labeling in order to illustrate your definition.
In our upcoming study of integral calculus for multivariable functions, we will first extend the idea of the single-variable definite integral to functions of two variables over rectangular domains. To do so, we will need to understand how to partition a rectangle into subrectangles. Let be rectangular domain (we can also represent this domain with the notation ), as pictured in Figure 11.1.1.
Figure11.1.1.Rectangular domain with subrectangles.
To form a partition of the full rectangular region, , we will partition both intervals and ; in particular, we choose to partition the interval into three uniformly sized subintervals and the interval into two evenly sized subintervals as shown in Figure 11.1.1. In the following questions, we discuss how to identify the endpoints of each subinterval and the resulting subrectangles.
Let be the endpoints of the subintervals of after partitioning. What is the length of each subinterval for from 1 to 3?
Explicitly identify ,,, and . On Figure 11.1.1 or your own version of the diagram, label these endpoints.
Let be the endpoints of the subintervals of after partitioning. What is the length of each subinterval for from 1 to 2? Identify ,, and and label these endpoints on Figure 11.1.1.
Let denote the subrectangle . Appropriately label each subrectangle in your drawing of Figure 11.1.1. How does the total number of subrectangles depend on the partitions of the intervals and ?
Subsection11.1.1Double Riemann Sums over Rectangles
For the definite integral in single-variable calculus, we considered a continuous function over a closed, bounded interval . In multivariable calculus, we will eventually develop the idea of a definite integral over a closed, bounded region (such as the interior of a circle). We begin with a simpler situation by thinking only about rectangular domains, and will address more complicated domains in Section 11.3.
Let be a continuous function defined on a rectangular domain . As we saw in Preview Activity 11.1.1, the domain is a rectangle and we want to partition into subrectangles. We do this by partitioning each of the intervals and into subintervals and using those subintervals to create a partition of into subrectangles. In the first activity, we address the quantities and notations we will use in order to define double Riemann sums and double integrals.
Let be defined on the rectangular domain . Partition the interval into four uniformly sized subintervals and the interval into three evenly sized subintervals as shown in Figure 11.1.2. As we did in Preview Activity 11.1.1, we will need a method for identifying the endpoints of each subinterval and the resulting subrectangles.
Figure11.1.2.Rectangular domain with subrectangles.
Let be the endpoints of the subintervals of after partitioning. Label these endpoints in Figure 11.1.2.
What is the length of each subinterval ? Your answer should be in terms of and .
Let be the endpoints of the subintervals of after partitioning. Label these endpoints in Figure 11.1.2.
What is the length of each subinterval ? Your answer should be in terms of and .
The partitions of the intervals and partition the rectangle into subrectangles. How many subrectangles are there?
Let denote the subrectangle . Label each subrectangle in Figure 11.1.2.
What is area of each subrectangle?
Now let and . Let be the point in the upper right corner of the subrectangle . Identify and correctly label this point in Figure 11.1.2. Calculate the product
Explain, geometrically, what this product represents.
For each and , choose a point in the subrectangle . Identify and correctly label these points in Figure 11.1.2. Explain what the product
represents.
If we were to add all the values for each and , what does the resulting number approximate about the surface defined by on the domain ? (You don’t actually need to add these values.)
Write a double sum using summation notation that expresses the arbitrary sum from part ().
Partition the interval into subintervals of equal length . Let ,,, be the endpoints of these subintervals, where .
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Partition the interval into subintervals of equal length . Let ,,, be the endpoints of these subintervals, where .
-.
These two partitions create a partition of the rectangle into subrectangles with opposite vertices and for between and and between and . These rectangles all have equal area .
Choose a point in each rectangle . Then, a double Riemann sum for over is given by
If on the rectangle , we may ask to find the volume of the solid bounded above by over , as illustrated on the left of Figure 11.1.4. This volume is approximated by a Riemann sum, which sums the volumes of the rectangular boxes shown on the right of Figure 11.1.4.
As we let the number of subrectangles increase without bound (in other words, as both and in a double Riemann sum go to infinity), as illustrated in Figure 11.1.5, the sum of the volumes of the rectangular boxes approaches the volume of the solid bounded above by over . The value of this limit, provided it exists, is the double integral.
Let be a rectangular region in the -plane and a continuous function over . With terms defined as in a double Riemann sum, the double integral of over is
Suppose that assumes both positive and negatives values on the rectangle , as shown on the left of Figure 11.1.7. When constructing a Riemann sum, for each and , the product can be interpreted as a “signed” volume of a box with base area and “signed” height . Since can have negative values, this “height” could be negative. The sum
can then be interpreted as a sum of “signed” volumes of boxes, with a negative sign attached to those boxes whose heights are below the -plane.
Figure11.1.7.The integral measures signed volume. We can then realize the double integral as a difference in volumes: tells us the volume of the solids the graph of bounds above the -plane over the rectangle minus the volume of the solids the graph of bounds below the -plane under the rectangle . This is shown on the right of Figure 11.1.7.
The average of the finitely many values that we take in a double Riemann sum is given by
Avg
If we take the limit as and go to infinity, we obtain what we define as the average value of over the region , which is connected to the value of the double integral. First, to view Avg as a double Riemann sum, note that
and
Thus,
where denotes the area of the rectangle . Then, the average value of the function over ,, is given by
Therefore, the double integral of over divided by the area of gives us the average value of the function on . Finally, if on , we can interpret this average value of on as the height of the box with base that has the same volume as the volume of the surface defined by over .
Draw a picture of . Partition into 2 subintervals of equal length and the interval into two subintervals of equal length. Draw these partitions on your picture of and label the resulting subrectangles using the labeling scheme we established in the definition of a double Riemann sum.
For each and , let be the midpoint of the rectangle . Identify the coordinates of each . Draw these points on your picture of .
Calculate the Riemann sum
using the partitions we have described. If we let be the midpoint of the rectangle for each and , then the resulting Riemann sum is called a midpoint sum.
Give two interpretations for the meaning of the sum you just calculated.
Let on the rectangular domain . Partition into 3 equal length subintervals and into 2 equal length subintervals. A table of values of at some points in is given in Table 11.1.8, and a graph of with the indicated partitions is shown in Figure 11.1.9.
Sketch the region in the plane using the values in Table 11.1.8 as the partitions.
Calculate the double Riemann sum using the given partition of and the values of in the upper right corner of each subrectangle.
Use geometry to calculate the exact value of and compare it to your approximation. Describe one way we could obtain a better approximation using the given data.
We conclude this section with a list of properties of double integrals. Since similar properties are satisfied by single-variable integrals and the arguments for double integrals are essentially the same, we omit their justification.
Let be a continuous function on a rectangle . The double Riemann sum for over is created as follows.
-.
Partition the interval into subintervals of equal length . Let ,,, be the endpoints of these subintervals, where .
-.
Partition the interval into subintervals of equal length . Let ,,, be the endpoints of these subintervals, where .
-.
These two partitions create a partition of the rectangle into subrectangles with opposite vertices and for between and and between and . These rectangles all have equal area .
-.
Choose a point in each rectangle . Then a double Riemann sum for over is given by
With terms defined as in the Double Riemann Sum, the double integral of over is
Two interpretations of the double integral are:
-.
The volume of the solids the graph of bounds above the -plane over the rectangle minus the volume of the solids the graph of bounds below the -plane under the rectangle ;
-.
Dividing the double integral of over by the area of gives us the average value of the function on . If on , we can interpret this average value of on as the height of the box with base that has the same volume as the volume of the surface defined by over .
Suppose and is the rectangle with vertices (0,0), (6,0), (6,4), (0,4). In each part, estimate using Riemann sums. For underestimates or overestimates, consistently use either the lower left-hand corner or the upper right-hand corner of each rectangle in a subdivision, as appropriate.
(b) Estimate three ways: by partitioning into four subrectangles and evaluating at its maximum and minimum values on each subrectangle, and then by considering the average of these (over and under) estimates.
Values of are given in the table below. Let be the rectangle . Find a Riemann sum which is a reasonable estimate for with and . Note that the values given in the table correspond to midpoints.
The temperature at any point on a metal plate in the plane is given by , where and are measured in inches and in degrees Celsius. Consider the portion of the plate that lies on the rectangular region .
Estimate the value of by using a double Riemann sum with two subintervals in each direction and choosing to be the point that lies in the upper right corner of each subrectangle.
Determine the area of the rectangle .
Estimate the average temperature, , over the region .
Do you think your estimate in (c) is an over- or under-estimate of the true temperature? Why?
Let be a function of independent variables and that is increasing in both the positive and directions on a rectangular domain . For each of the following situations, determine if the double Riemann sum of over is an overestimate or underestimate of the double integral , or if it impossible to determine definitively. Provide justification for your responses.
The wind chill, as frequently reported, is a measure of how cold it feels outside when the wind is blowing. In Table 11.1.10, the wind chill , measured in degrees Fahrenheit, is a function of the wind speed , measured in miles per hour, and the ambient air temperature , also measured in degrees Fahrenheit. Approximate the average wind chill on the rectangle using 3 subintervals in the direction, 4 subintervals in the direction, and the point in the lower left corner in each subrectangle.
Estimate the value of by using a double Riemann sum with four subintervals in the direction and three subintervals in the direction, and choosing to be the point that is the midpoint of each subrectangle.
What important quantity does your double Riemann sum in (a) estimate?
Suppose it can be determined that . What is the exact average value of over ?
If you wanted to build a rectangular box (with the same base) that has the same volume as the box with the sloped top described here, how tall would the rectangular box have to be?