Subsection 5.1.1 Double Integrals over Rectangles
The simplest extension of the idea of the definite integral \(\ds\int_a^b f(x)\ dx\) over interval \(a \leq x \leq y\) is to try to "combine" or "integrate" all values of a function \(f(x,y)\) for arguments \(a \leq x \leq b\text{,}\) \(c \leq y \leq d\text{.}\) That is, we consider the function on a rectangular domain
\begin{equation*}
R = [a,b] \times [c,d] = \{(x,y) \in \mathbb{R}^2 : a \leq x \leq b,\ c \leq y \leq d \}
\end{equation*}
which has the same role as the interval \(I=[a,b]\) for the definite integral \(\int_a^b f(x)\ dx\text{.}\)
Subsubsection 5.1.1.1 The Definite Integral and Area Under the Graph of \(f(x)\)
Recall the definition
\begin{equation*}
\int_a^b f(x)\ dx = \lim_{n \to \infty} \sum_{i=1}^n f(x_i^*) \Delta x
\end{equation*}
where the points \(x_i^*\) each lie in one of the \(n\) subintervals of interval \([a,b]\) of width \(\Delta x = (b-a)/n\text{,}\) and recall that this is motivated as a calculation of the area between the curve \(y=f(x)\) and the \(x\)-axis over the interval \([a,b]\text{,}\) at least if \(f(x) >0\text{,}\) got from approximations with the sums of the area of many thin rectangles.
Likewise we will motivate the definition of the definite integral of \(f(x,y)\) over region \(R\) as the volume of the solid region between the surface \(z=f(x,y)\) and the \(x\)-\(y\) plane over this rectangle, and start with approximations with the sums of the volume of many thin thin rectangular boxes ("matchsticks"), each with base a small rectangle and height given by \(f\text{.}\)
Subsubsection 5.1.1.2 Double Integrals and Volume under the Graph of \(f(x,y)\)
As before, the interval \(a \leq x \leq b\) is divided into \(n\) equal subintervals \([x_{i-1},x_i]\text{,}\) \(x_i=a+i\Delta x\text{,}\) \(\Delta x = (b-a)/n\text{.}\) Likewise we divide interval \(c \leq y \leq d\) is divided into \(m\) equal subintervals \([y_{j-1},y_j]\text{,}\) \(y_j=a+j\Delta y\text{,}\) \(\Delta y = (d-c)/m\text{.}\) This divides the region \(R\) into \(nm\) little rectangles each of area \(\Delta A = \Delta x \Delta y\text{,}\) and in each of these we choose a point \((x_{ij}^*,y_{ij}^*)\) where \(f\) will be evaluated, to give the height of a thin rectangular box.
The volume of each matchstick is \(f(x_{ij}^*,y_{ij}^*)\Delta A\text{,}\) giving approximate volume for the whole solid as
\begin{equation*}
V \approx \sum_{i=1}^n \sum_{j=1}^m f(x_{ij}^*,y_{ij}^*)\Delta A.
\end{equation*}
Subsubsection 5.1.1.3 The Midpoint Rule
Perhaps most natural choice of these points is to put each one at the middle of the rectangle:
\begin{align*}
x_{ij}^* \amp= \bar{x}_i = \frac{x_{i-1}+x_i}{2} = a+(i-1/2)\Delta x,\\
y_{ij}^* \amp= \bar{y}_j = \frac{y_{j-1}+y_j}{2} = c+(j-1/2)\Delta y,
\end{align*}
so that
\begin{equation*}
V \approx \sum_{i=1}^n \sum_{j=1}^m f( \bar{x}_i ,\bar{y}_j)\Delta A.
\end{equation*}
This is indeed useful if you wish to approximate volumes numerically.
Subsubsection 5.1.1.4 Exact Volume as a Limit
Given the above approximation of the volume, it is natural to define the exact volume as the limit when the rectangles get ever smaller:
\begin{equation*}
V = \lim_{m,n \to \infty} \sum_{i=1}^n \sum_{j=1}^m f(x_{ij}^*,y_{ij}^*)\Delta A.
\end{equation*}
Subsubsection 5.1.1.5 The Double Integral
The above idea is also useful in other contexts where the value of \(f\) need not be non-negative, so more generally we use:
Definition 5.1.1.
For function \(f\) continuous on rectangle \(R\text{,}\) the double integral of \(f\) over rectangle \(R\) is
\begin{equation*}
\iint\limits_R f(x,y)\ dA = \lim_{m,n \to \infty}\sum_{i=1}^n \sum_{j=1}^m f(x_{ij}^*,y_{ij}^*)\Delta A
\end{equation*}
This appears to depend on the choice of the values \(x_{ij}^*\) and \(y_{ij}^*\) for each choice of \(n\) and \(m\text{.}\) However, it can be proven that due to continuity, the result is the same for any choice.
In fact more is true: the spacing of the points \(x_i\) and \(y_j\) does not have to be equal, and the evaluation points do not need to be the midpoints, they just need to lie in the relevant small rectangle: \((x^*_{ij}, y^*_{ij}) \in [x_{i-1}, x_i] \times [y_{j-1}, y_j]\text{.}\)
What still matters is that all the intervals shrink to length zero: with
\begin{align*}
\Delta x_i \amp= x_i-x_{i-1}, \amp \Delta x = \max\limits_i \Delta x_i\\
\Delta y_j \amp= y_j-y_{j-1}, \amp \Delta y = \max\limits_j \Delta y_j\\
\Delta A_{ij} \amp = \Delta x_i \Delta y_j
\end{align*}
one can say that so when \(\Delta x \to 0\) and \(\Delta y \to 0\text{,}\)
\begin{equation*}
\sum_{i=1}^n \sum_{j=1}^m f(x_{ij}^*,y_{ij}^*) \Delta A_{ij} \to \iint\limits_R f(x,y)\ dA.
\end{equation*}
The sum appearing in this definition,
\begin{equation*}
\sum_{i=1}^n \sum_{j=1}^m f(x_{ij}^*,y_{ij}^*) \Delta A_{ij}
\end{equation*}
is called a double Riemann sum, and includes the volume approximation above as an example. Thus it is natural to define the volume of the solid over rectangle \(R\) and under the graph of a positive-valued function \(f\) to be the value of this double integral.
We will see in
Subsection 5.1.3 how such integrals can sometimes be evaluated using results we already know for integrals of functions of one variable.
Subsection 5.1.2 Properties of Double Integrals Over Rectangles
Before learning how to evaluate such integrals, we note a few rather intuitive and familiar properties about sums, constant multiples and comparisons.
\begin{equation}
\iint\limits_R f(x,y) + g(x,y) \ dA = \iint\limits_R f(x,y) \ dA + \iint\limits_R g(x,y) \ dA.\tag{5.1.1}
\end{equation}
\begin{equation}
\iint\limits_R k f(x,y) ) \ dA = k \iint\limits_R f(x,y) \ dA \text{ for any } k.\tag{5.1.2}
\end{equation}
\begin{equation}
\text{ If }
f(x,y) \leq g(x,y)\text{ then }
\iint\limits_R f(x,y) ) \ dA \leq \iint\limits_R g(x,y) \ dA\tag{5.1.3}
\end{equation}
when the first inequality holds for all points \((x,y)\) in \(R\text{.}\)
Once we define double integrals over more general domains
\(D \subset \reals^2\) in
Section 5.2 it will be noted in
Subsection 5.2.4 that these properties all still hold.
Subsection 5.1.3 Iterated Integrals and Fubini’s Theorem
The essential idea of this section is one formula, which in a sense does for double integrals what the Fundamental Theorems of Calculus did for definite integrals, by allowing evaluation using anti-derivatives.
Theorem 5.1.2. (Fubini).
If \(f\) is continuous on the rectangle \(R=[a,b] \times [c,d]\) then
\begin{equation*}
\iint\limits_R f(x,y)\ dA
= \int_{x=a}^b \left[ \int_{y=c}^d f(x,y)\ dy \right] dx
= \int_{y=c}^d \left[ \int_{x=a}^b f(x,y)\ dx \right] dy
\end{equation*}
In particular, all the integrals here exist.
More generally, if \(f\) is bounded on \(R\) and is continuous except except on a finite number of smooth curves, all these integrals exist and are equal.
(Aside: Indicating the variable name for each integral with "\(\int_{x=a}^b\)" and "\(\int_{y=c}^d\)" is not mandatory, but it can help with clarity, so I recommend it. It also follows the pattern of summation notation, "\(\sum_{i=1}^n\)".)
The allowance for being discontinuous along curves will be important in the next section!
The full proof of Fubini’s Theorem is not given here; it is dealt with in a more advanced calculus course (such as Math 311: Advanced Calculus). However, it is fairly easy to verify in the special case of \(\iint_R f(x)g(y) dA\text{,}\) as will be seen soon.
The main objective here is to explain iterated integrals and see how to evaluate them using familiar techniques for definite integrals.
An iterated integral is one like the middle expression above:
\begin{equation*}
\int_{x=a}^b \left[ \int_{y=c}^d f(x,y)\ dy \right] dx
\end{equation*}
(which can be shortened as \(\int_a^b \int_c^d f(x,y)\ dy\ dx\)).
Evaluation of this iterated integral starts with the inside integral
\begin{equation*}
\int_{y=c}^d f(x,y)\ dy
\end{equation*}
which is what is sometimes called a partial integral: like a partial derivative, we focus on the variable \(y\) that appears in the differential \(dy\) (and sometime in the limits of integration, as a reminder!), and treat \(x\) as a constant.
Subsubsection 5.1.3.1 Evaluating Partial Integrals
For any given \(x\) value (in the range \(a \leq x \leq b\)) this definite integral gives a number, but since that number depends on \(x\text{,}\) the overall result is a function of \(x\text{:}\) \(\ds\int_{y=c}^d f(x,y)\ dy = G(x)\text{,}\) and so
\begin{equation*}
\int_{x=a}^b \int_{y=c}^d f(x,y)\ dy\ dx
= \int_{a}^b G(x)\ dx
\end{equation*}
So once the partial integral is evaluated, what remains is a familiar definite integral in variable \(x\text{.}\)
This then gives a number (not a function of \(x\) or \(y\)), which by Fubini’s Theorem is also the value of the double integral.
Subsubsection 5.1.3.2 A Product Rule
One useful special case is when the integrand is a product of two functions, one in each variable:
\begin{equation*}
\iint\limits_R f(x)g(y)\ dA = \int_{x=a}^b \int_{y=c}^d f(x)g(y)\ dy\ dx
\end{equation*}
In the inner integral, \(x\) is effectively a constant, so \(f(x)\) is a constant factor, and can be moved outside of this inner, \(y\) integral:
\begin{equation*}
\int_{x=a}^b \int_{y=c}^d f(x)g(y)\ dy\ dx = \int_{x=a}^b f(x) \left[ \int_{y=c}^d g(y)\ dy\right] dx
\end{equation*}
Now, the whole integral \(\int_{y=c}^d g(y)\ dy\) gives a constant, so can be moved outside the outer \(x\) integral:
\begin{equation*}
\int_{x=a}^b f(x) \left[ \int_{y=c}^d g(y)\ dy \right] dx = \int_{y=c}^d g(y)\ dy \int_{x=a}^b f(x) \ dx
\end{equation*}
All together, the double integral of a product has been broken into a product of "single" integrals:
Proposition 5.1.3. A Product Rule.
\begin{equation}
\iint\limits_R f(x)g(y)\ dA = \int_{x=a}^b \int_{y=c}^d f(x)g(y)\ dy\ dx = \int_a^b f(x)\ dx \int_c^d g(y)\ dy\tag{5.1.4}
\end{equation}
Subsubsection 5.1.3.3 Iterated Integrals Over Other Regions
In an iterated integral, the dummy variable of the outer integral is fixed in the inner integral: that is, the inner integral is evaluated once for each value of the "outer" variable.
Thus, it makes sense for the limits of the inner integral to depend on the that outer variable. This leads to integral of the forms
\begin{equation*}
\int_{x=a}^{x=b}\int_{y=B(x)}^{y=T(x)} f(x,y)\ dy\ dx
\quad \mbox{and} \quad
\int_{y=c}^{y=d}\int_{x=L(y)}^{x=R(y)} f(x,y)\ dx\ dy.
\end{equation*}
In the first of these, the set of all \(x\) and \(y\) values in \(\reals^2\) involved are those between the vertical lines \(x=a\) and \(x=b\text{,}\) bounded above and below by a top curve \(y=T(x)\) and a bottom curve \(y=B(x)\text{,}\) and the inner integral gives
\begin{equation*}
G(x) = \int_{y=B(x)}^{y=T(x)} f(x,y)\ dy\text{.}
\end{equation*}
Similarly, in the second, the set of all \(x\) and \(y\) values are those between the horizontal lines \(y=c\) and \(y=d\text{,}\) bounded at the sides by a left curve \(x=L(y)\) and a right curve \(x=R(y)\text{.}\)
Such integrals will be part of our strategy for evaluating integrals over more general regions.
