Coordinated Multivariable Calculus

Subsection3.1.1Double 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 Section3.3.

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 Activity3.1.1, 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.

Activity3.1.2

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 Figure3.1.2. As we did in Preview Activity3.1.1, we will need a method for identifying the endpoints of each subinterval and the resulting subrectangles.

1. 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 Figure3.1.2.

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 \(b\text{.}\)

3. 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 Figure3.1.2.

4. What is the length \(\Delta y\) of each subinterval \([y_{j-1},y_j]\text{?}\) Your answer should be in terms of \(c\) and \(d\text{.}\)

5. The partitions of the intervals \([a,b]\) and \([c,d]\) partition the rectangle \(R\) into subrectangles. How many subrectangles are there?

6. Let \(R_{ij}\) denote the subrectangle \([x_{i-1},x_i] \times [y_{j-1},y_j]\text{.}\) Label each subrectangle in Figure3.1.2.

7. What is area \(\Delta A\) of each subrectangle?

8. 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 Figure3.1.2. Calculate the product

   \begin{equation*} f(x_{11}^*,y_{11}^*) \Delta A. \end{equation*}

   Explain, geometrically, what this product represents.

9. 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 Figure3.1.2. Explain what the product

   \begin{equation*} f(x_{ij}^*,y_{ij}^*) \Delta A \end{equation*}

   represents.

10. 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.)

11. Write a double sum using summation notation that expresses the arbitrary sum from part (\(j\)).

Subsection3.1.2Double Riemann Sums and Double Integrals

Now we use the process from the most recent activity to formally define double Riemann sums and double integrals.

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 Figure3.1.4. This volume is approximated by a Riemann sum, which sums the volumes of the rectangular boxes shown on the right of Figure3.1.4.

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 Figure3.1.5, 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.

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.

Subsection3.1.3Interpretation 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 Figure3.1.7. 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

  \begin{equation*} \sum_{j=1}^n \sum_{i=1}^m f(x_{ij}^*, y_{ij}^*) \cdot \Delta A \end{equation*}

  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.

  

  

  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 Figure3.1.7.

• The average of the finitely many \(mn\) values \(f(x_{ij}^*, y_{ij}^*)\) that we take in a double Riemann sum is given by

  \begin{equation*} \mbox{Avg} _{mn} = \frac{1}{mn} \sum_{j=1}^n \sum_{i=1}^m f(x_{ij}^*, y_{ij}^*). \end{equation*}

  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

  \begin{equation*} \Delta x = \frac{b-a}{m} \ \ \ \ \ \text{ and } \ \ \ \ \ \Delta y = \frac{d-c}{n}. \end{equation*}

  Thus,

  \begin{equation*} \frac{1}{mn} = \frac{\Delta x \cdot \Delta y}{(b-a)(d-c)} = \frac{\Delta A}{A(R)}, \end{equation*}

  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

  \begin{align*} f_{\operatorname{AVG}(R)} \amp = \lim_{m,n \to \infty} \frac{1}{mn} \sum_{j=1}^n \sum_{i=1}^m f(x_{ij}^*, y_{ij}^*)\\ \amp = \lim_{m,n \to \infty} \frac{1}{A(R)} \sum_{j=1}^n \sum_{i=1}^m f(x_{ij}^*, y_{ij}^*) \cdot \Delta A\\ \amp = \frac{1}{A(R)} \iint_R f(x,y) \, dA. \end{align*}

  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 \(R\text{.}\)

Activity3.1.4

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 Table3.1.8, and a graph of \(f\) with the indicated partitions is shown in Figure3.1.9.

	\(-2\)	\(-1\)	\(0\)	\(1\)	\(2\)
\(1\)	\(0\)	\(\sqrt{3}\)	\(2\)	\(\sqrt{3}\)	\(0\)
\(2\)	\(0\)	\(\sqrt{3}\)	\(2\)	\(\sqrt{3}\)	\(0\)
\(3\)	\(0\)	\(\sqrt{3}\)	\(2\)	\(\sqrt{3}\)	\(0\)
\(4\)	\(0\)	\(\sqrt{3}\)	\(2\)	\(\sqrt{3}\)	\(0\)
\(5\)	\(0\)	\(\sqrt{3}\)	\(2\)	\(\sqrt{3}\)	\(0\)
\(6\)	\(0\)	\(\sqrt{3}\)	\(2\)	\(\sqrt{3}\)	\(0\)
\(7\)	\(0\)	\(\sqrt{3}\)	\(2\)	\(\sqrt{3}\)	\(0\)

1. Sketch the region \(R\) in the plane using the values in Table3.1.8 as the partitions.

2. Calculate the double Riemann sum using the given partition of \(R\) and the values of \(f\) in the upper right corner of each subrectangle.

3. 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.

Subsection3.1.4Summary

• 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

    \begin{equation*} \sum_{j=1}^n \sum_{i=1}^m f(x_{ij}^*, y_{ij}^*) \cdot \Delta A. \end{equation*}

• With terms defined as in the Double Riemann Sum, the double integral of \(f\) over \(R\) is

  \begin{equation*} \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. \end{equation*}

• 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 \(R\text{.}\)