How is the double integral of a continuous function \(f = f(x,y)\) defined?
What are two things the double integral of a function can tell us?
In single-variable calculus, recall that we approximated the area under the graph of a positive function \(f\) on an interval \([a,b]\) by adding areas of rectangles whose heights are determined by the curve. The general process involved subdividing the interval \([a,b]\) 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.
Preview Activity \(\PageIndex{1}\)
In this activity we introduce the concept of a double Riemann sum.
Review the concept of the Riemann sum from single-variable calculus. Then, explain how we define the definite integral \(\int_a^b f(x) \ dx\) of a continuous function of a single variable \(x\) on an interval \([a,b]\text{.}\) Include a sketch of a continuous function on an interval \([a,b]\) 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 \(R\) be rectangular domain \(R = \{(x,y) : 0 \leq x \leq 6, 2 \leq y \leq 4\}\) (we can also represent this domain with the notation \([0,6] \times [2,4]\)), as pictured in Figure \(\PageIndex{1}\).
Figure \(\PageIndex{1}\). Rectangular domain \(R\) with subrectangles.
To form a partition of the full rectangular region, \(R\text{,}\) we will partition both intervals \([0,6]\) and \([2,4]\text{;}\) in particular, we choose to partition the interval \([0,6]\) into three uniformly sized subintervals and the interval \([2,4]\) into two evenly sized subintervals as shown in Figure \(\PageIndex{1}\). In the following questions, we discuss how to identify the endpoints of each subinterval and the resulting subrectangles.
Let \(0=x_0 \lt x_1 \lt x_2 \lt x_3=6\) be the endpoints of the subintervals of \([0,6]\) after partitioning. What is the length \(\Delta x\) of each subinterval \([x_{i-1},x_i]\) for \(i\) from 1 to 3?
Explicitly identify \(x_0\text{,}\) \(x_1\text{,}\) \(x_2\text{,}\) and \(x_3\text{.}\) On Figure \(\PageIndex{1}\) or your own version of the diagram, label these endpoints.
Let \(2=y_0 \lt y_1 \lt y_2=4\) be the endpoints of the subintervals of \([2,4]\) after partitioning. What is the length \(\Delta y\) of each subinterval \([y_{j-1},y_j]\) for \(j\) from 1 to 2? Identify \(y_0\text{,}\) \(y_1\text{,}\) and \(y_2\) and label these endpoints on Figure \(\PageIndex{1}\).
Let \(R_{ij}\) denote the subrectangle \([x_{i-1},x_i] \times [y_{j-1},y_j]\text{.}\) Appropriately label each subrectangle in your drawing of Figure \(\PageIndex{1}\). How does the total number of subrectangles depend on the partitions of the intervals \([0,6]\) and \([2,4]\text{?}\)
What is area \(\Delta A\) of each subrectangle?
Double Riemann Sums over Rectangles
For the definite integral in single-variable calculus, we considered a continuous function over a closed, bounded interval \([a,b]\text{.}\) 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 \(f = f(x,y)\) be a continuous function defined on a rectangular domain \(R = \{(x,y) : a \leq x \leq b, c \leq y \leq d\}\text{.}\) As we saw in Preview Activity \(\PageIndex{1}\), the domain is a rectangle \(R\) and we want to partition \(R\) into subrectangles. We do this by partitioning each of the intervals \([a,b]\) and \([c,d]\) into subintervals and using those subintervals to create a partition of \(R\) 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.
Activity \(\PageIndex{2}\)
Let \(f(x,y) = 100 - x^2-y^2\) be defined on the rectangular domain \(R = [a,b] \times [c,d]\text{.}\) Partition the interval \([a,b]\) into four uniformly sized subintervals and the interval \([c,d]\) into three evenly sized subintervals as shown in Figure \(\PageIndex{2}\). As we did in Preview Activity \(\PageIndex{1}\), we will need a method for identifying the endpoints of each subinterval and the resulting subrectangles.
Figure \(\PageIndex{2}\). Rectangular domain with subrectangles.
Let \(a=x_0 \lt x_1 \lt x_2 \lt x_3 \lt x_4 =b\) be the endpoints of the subintervals of \([a,b]\) after partitioning. Label these endpoints in Figure \(\PageIndex{2}\).
What is the length \(\Delta x\) of each subinterval \([x_{i-1},x_i]\text{?}\) Your answer should be in terms of \(a\) and \(b\text{.}\)
Let \(c=y_0 \lt y_1 \lt y_2 \lt y_3 =d\) be the endpoints of the subintervals of \([c,d]\) after partitioning. Label these endpoints in Figure \(\PageIndex{2}\).
What is the length \(\Delta y\) of each subinterval \([y_{j-1},y_j]\text{?}\) Your answer should be in terms of \(c\) and \(d\text{.}\)
The partitions of the intervals \([a,b]\) and \([c,d]\) partition the rectangle \(R\) into subrectangles. How many subrectangles are there?
Let \(R_{ij}\) denote the subrectangle \([x_{i-1},x_i] \times [y_{j-1},y_j]\text{.}\) Label each subrectangle in Figure \(\PageIndex{2}\).
What is area \(\Delta A\) of each subrectangle?
Now let \([a,b] = [0,8]\) and \([c,d] = [2,6]\text{.}\) Let \((x_{11}^*,y_{11}^*)\) be the point in the upper right corner of the subrectangle \(R_{11}\text{.}\) Identify and correctly label this point in Figure \(\PageIndex{2}\). Calculate the product
\[ f(x_{11}^*,y_{11}^*) \Delta A. \nonumber \]
Explain, geometrically, what this product represents.
For each \(i\) and \(j\text{,}\) choose a point \((x_{ij}^*,y_{ij}^*)\) in the subrectangle \(R_{i,j}\text{.}\) Identify and correctly label these points in Figure \(\PageIndex{2}\). Explain what the product
\[ f(x_{ij}^*,y_{ij}^*) \Delta A \nonumber \]
represents.
If we were to add all the values \(f(x_{ij}^*,y_{ij}^*) \Delta A\) for each \(i\) and \(j\text{,}\) what does the resulting number approximate about the surface defined by \(f\) on the domain \(R\text{?}\) (You don't actually need to add these values.)
Write a double sum using summation notation that expresses the arbitrary sum from part (\(j\)).
Double Riemann Sums and Double Integrals
Now we use the process from the most recent activity to formally define double Riemann sums and double integrals.
Definition \(\PageIndex{1}\)
Let \(f\) be a continuous function on a rectangle \(R = \{(x,y) : a \leq x \leq b, c \leq y \leq d\}\text{.}\) A double Riemann sum for \(f\) over \(R\) is created as follows.
Partition the interval \([a, b]\) into \(m\) subintervals of equal length \(\Delta x = \frac{b-a}{m}\text{.}\) Let \(x_0\text{,}\) \(x_1\text{,}\) \(\ldots\text{,}\) \(x_m\) be the endpoints of these subintervals, where \(a = x_0\lt x_1\lt x_2 \lt \cdots \lt x_m = b\text{.}\)
Partition the interval \([c, d]\) into \(n\) subintervals of equal length \(\Delta y = \frac{d-c}{n}\text{.}\) Let \(y_0\text{,}\) \(y_1\text{,}\) \(\ldots\text{,}\) \(y_n\) be the endpoints of these subintervals, where \(c = y_0\lt y_1\lt y_2 \lt \cdots \lt y_n = d\text{.}\)
These two partitions create a partition of the rectangle \(R\) into \(mn\) subrectangles \(R_{ij}\) with opposite vertices \((x_{i-1},y_{j-1})\) and \((x_i, y_j)\) for \(i\) between \(1\) and \(m\) and \(j\) between \(1\) and \(n\text{.}\) These rectangles all have equal area \(\Delta A = \Delta x \cdot \Delta y\text{.}\)
Choose a point \((x_{ij}^*, y_{ij}^*)\) in each rectangle \(R_{ij}\text{.}\) Then, a double Riemann sum for \(f\) over \(R\) is given by
\[ \sum_{j=1}^n \sum_{i=1}^m f(x_{ij}^*, y_{ij}^*) \cdot \Delta A. \nonumber \]
If \(f(x,y) \geq 0\) on the rectangle \(R\text{,}\) we may ask to find the volume of the solid bounded above by \(f\) over \(R\text{,}\) as illustrated on the left of Figure \(\PageIndex{3}\). This volume is approximated by a Riemann sum, which sums the volumes of the rectangular boxes shown on the right of Figure \(\PageIndex{3}\).
Figure \(\PageIndex{3}\). The volume under a graph approximated by a Riemann Sum.
As we let the number of subrectangles increase without bound (in other words, as both \(m\) and \(n\) in a double Riemann sum go to infinity), as illustrated in Figure \(\PageIndex{4}\), the sum of the volumes of the rectangular boxes approaches the volume of the solid bounded above by \(f\) over \(R\text{.}\) The value of this limit, provided it exists, is the double integral.
Figure \(\PageIndex{4}\). Finding better approximations by using smaller subrectangles.
Definition \(\PageIndex{2}\)
Let \(R\) be a rectangular region in the \(xy\)-plane and \(f\) a continuous function over \(R\text{.}\) With terms defined as in a double Riemann sum, the double integral of \(f\) over \(R\) is
\[ \iint_R f(x,y) \, dA = \lim_{m,n \to \infty} \sum_{j=1}^n \sum_{i=1}^m f(x_{ij}^*, y_{ij}^*) \cdot \Delta A. \nonumber \]
Some textbooks use the notation \(\int_R f(x,y) \, dA\) for a double integral. You will see this in some of the WeBWorK problems.
Interpretation of Double Riemann Sums and Double integrals.
At the moment, there are two ways we can interpret the value of the double integral.
Suppose that \(f(x,y)\) assumes both positive and negatives values on the rectangle \(R\text{,}\) as shown on the left of Figure \(\PageIndex{5}\). When constructing a Riemann sum, for each \(i\) and \(j\text{,}\) the product \(f(x_{ij}^*, y_{ij}^*) \cdot \Delta A\) can be interpreted as a “signed” volume of a box with base area \(\Delta A\) and “signed” height \(f(x_{ij}^*, y_{ij}^*)\text{.}\) Since \(f\) can have negative values, this “height” could be negative. The sum
\[ \sum_{j=1}^n \sum_{i=1}^m f(x_{ij}^*, y_{ij}^*) \cdot \Delta A \nonumber \]
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 \(xy\)-plane.
Figure \(\PageIndex{5}\). The integral measures signed volume.
We can then realize the double integral \(\iint_R f(x,y) \, dA\) as a difference in volumes: \(\iint_R f(x,y) \, dA\) tells us the volume of the solids the graph of \(f\) bounds above the \(xy\)-plane over the rectangle \(R\) minus the volume of the solids the graph of \(f\) bounds below the \(xy\)-plane under the rectangle \(R\text{.}\) This is shown on the right of Figure \(\PageIndex{5}\).
The average of the finitely many \(mn\) values \(f(x_{ij}^*, y_{ij}^*)\) that we take in a double Riemann sum is given by
If we take the limit as \(m\) and \(n\) go to infinity, we obtain what we define as the average value of \(f\) over the region \(R\text{,}\) which is connected to the value of the double integral. First, to view \(\text{ Avg } _{mn}\) as a double Riemann sum, note that
\[ \Delta x = \frac{b-a}{m} \ \ \ \ \ \text{ and } \ \ \ \ \ \Delta y = \frac{d-c}{n}. \nonumber \]
where \(A(R)\) denotes the area of the rectangle \(R\text{.}\) Then, the average value of the function \(f\) over \(R\text{,}\) \(f_{\operatorname{AVG}(R)}\text{,}\) is given by
Therefore, the double integral of \(f\) over \(R\) divided by the area of \(R\) gives us the average value of the function \(f\) on \(R\text{.}\) Finally, if \(f(x, y) \geq 0\) on \(R\text{,}\) we can interpret this average value of \(f\) on \(R\) as the height of the box with base \(R\) that has the same volume as the volume of the surface defined by \(f\) over \(R\text{.}\)
Activity \(\PageIndex{3}\)
Let \(f(x,y) = x+2y\) and let \(R = [0,2] \times [1,3]\text{.}\)
Draw a picture of \(R\text{.}\) Partition \([0,2]\) into 2 subintervals of equal length and the interval \([1,3]\) into two subintervals of equal length. Draw these partitions on your picture of \(R\) and label the resulting subrectangles using the labeling scheme we established in the definition of a double Riemann sum.
For each \(i\) and \(j\text{,}\) let \((x_{ij}^*, y_{ij}^*)\) be the midpoint of the rectangle \(R_{ij}\text{.}\) Identify the coordinates of each \((x_{ij}^*, y_{ij}^*)\text{.}\) Draw these points on your picture of \(R\text{.}\)
Calculate the Riemann sum
\[ \sum_{j=1}^n \sum_{i=1}^m f(x_{ij}^*, y_{ij}^*) \cdot \Delta A \nonumber \]
using the partitions we have described. If we let \((x_{ij}^*, y_{ij}^*)\) be the midpoint of the rectangle \(R_{ij}\) for each \(i\) and \(j\text{,}\) then the resulting Riemann sum is called a midpoint sum.
Give two interpretations for the meaning of the sum you just calculated.
Activity \(\PageIndex{4}\)
Let \(f(x,y) = \sqrt{4-y^2}\) on the rectangular domain \(R = [1,7] \times [-2,2]\text{.}\) Partition \([1,7]\) into 3 equal length subintervals and \([-2,2]\) into 2 equal length subintervals. A table of values of \(f\) at some points in \(R\) is given in Table \(\PageIndex{1}\), and a graph of \(f\) with the indicated partitions is shown in Figure \(\PageIndex{6}\).
Table \(\PageIndex{1}\). Table of values of \(f(x,y) = \sqrt{4-y^2}\text{.}\)
\(-2\)
\(-1\)
\(0\)
\(1\)
\(2\)
\(1\)
\(0\)
\(\sqrt{3}\)
\(2\)
\(\sqrt{3}\)
\(0\)
\(2\)
\(0\)
\(\sqrt{3}\)
\(2\)
\(\sqrt{3}\)
\(0\)
\(3\)
\(0\)
\(\sqrt{3}\)
\(2\)
\(\sqrt{3}\)
\(0\)
\(4\)
\(0\)
\(\sqrt{3}\)
\(2\)
\(\sqrt{3}\)
\(0\)
\(5\)
\(0\)
\(\sqrt{3}\)
\(2\)
\(\sqrt{3}\)
\(0\)
\(6\)
\(0\)
\(\sqrt{3}\)
\(2\)
\(\sqrt{3}\)
\(0\)
\(7\)
\(0\)
\(\sqrt{3}\)
\(2\)
\(\sqrt{3}\)
\(0\)
Figure \(\PageIndex{6}\). Graph of \(f(x,y) = \sqrt{4-y^2}\) on \(R\text{.}\)
Sketch the region \(R\) in the plane using the values in Table \(\PageIndex{1}\) as the partitions.
Calculate the double Riemann sum using the given partition of \(R\) and the values of \(f\) in the upper right corner of each subrectangle.
Use geometry to calculate the exact value of \(\iint_R f(x,y) \, dA\) 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.
Properties of Double Integrals
Let \(f\) and \(g\) be continuous functions on a rectangle \(R = \{(x,y) : a \leq x \leq b, c \leq y \leq d\}\text{,}\) and let \(k\) be a constant. Then
\(\iint_R (f(x,y) + g(x,y)) \, dA = \iint_R f(x,y) \, dA + \iint_R g(x,y) \, dA\text{.}\)
\(\iint_R kf(x,y) \, dA = k \iint_R f(x,y) \, dA\text{.}\)
If \(f(x,y) \geq g(x,y)\) on \(R\text{,}\) then \(\iint_R f(x,y) \, dA \geq \iint_R g(x,y) \, dA\text{.}\)
Summary
Let \(f\) be a continuous function on a rectangle \(R = \{(x,y) : a \leq x \leq b, c \leq y \leq d\}\text{.}\) The double Riemann sum for \(f\) over \(R\) is created as follows.
Partition the interval \([a, b]\) into \(m\) subintervals of equal length \(\Delta x = \frac{b-a}{m}\text{.}\) Let \(x_0\text{,}\) \(x_1\text{,}\) \(\ldots\text{,}\) \(x_m\) be the endpoints of these subintervals, where \(a = x_0\lt x_1\lt x_2 \lt \cdots \lt x_m = b\text{.}\)
Partition the interval \([c, d]\) into \(n\) subintervals of equal length \(\Delta y = \frac{d-c}{n}\text{.}\) Let \(y_0\text{,}\) \(y_1\text{,}\) \(\ldots\text{,}\) \(y_n\) be the endpoints of these subintervals, where \(c = y_0\lt y_1\lt y_2 \lt \cdots \lt y_n = d\text{.}\)
These two partitions create a partition of the rectangle \(R\) into \(mn\) subrectangles \(R_{ij}\) with opposite vertices \((x_{i-1},y_{j-1})\) and \((x_i, y_j)\) for \(i\) between \(1\) and \(m\) and \(j\) between \(1\) and \(n\text{.}\) These rectangles all have equal area \(\Delta A = \Delta x \cdot \Delta y\text{.}\)
Choose a point \((x_{ij}^*, y_{ij}^*)\) in each rectangle \(R_{ij}\text{.}\) Then a double Riemann sum for \(f\) over \(R\) is given by
\[ \sum_{j=1}^n \sum_{i=1}^m f(x_{ij}^*, y_{ij}^*) \cdot \Delta A. \nonumber \]
With terms defined as in the Double Riemann Sum, the double integral of \(f\) over \(R\) is
\[ \iint_R f(x,y) \, dA = \lim_{m,n \to \infty} \sum_{j=1}^n \sum_{i=1}^m f(x_{ij}^*, y_{ij}^*) \cdot \Delta A. \nonumber \]
Two interpretations of the double integral \(\iint_R f(x,y) \, dA\) are:
The volume of the solids the graph of \(f\) bounds above the \(xy\)-plane over the rectangle \(R\) minus the volume of the solids the graph of \(f\) bounds below the \(xy\)-plane under the rectangle \(R\text{;}\)
Dividing the double integral of \(f\) over \(R\) by the area of \(R\) gives us the average value of the function \(f\) on \(R\text{.}\) If \(f(x, y) \geq 0\) on \(R\text{,}\) we can interpret this average value of \(f\) on \(R\) as the height of the box with base \(R\) that has the same volume as the volume of the surface defined by \(f\) over \(R\text{.}\)
