Motivating Questions
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How do we evaluate a double integral over a rectangle as an iterated integral, and why does this process work?
Section 12.2 Iterated Integrals
Subsection 12.2.1 Introduction
In Section 12.1, we defined the double integral of a continuous function \(f = f(x,y)\) over a rectangle \(R = [a,b] \times [c,d]\) as
\begin{equation*}
\iint_R f(x,y) \, dA = \lim_{m,n \to \infty} \sum_{j=1}^n \sum_{i=1}^m f\left(x_{ij}^*, y_{ij}^*\right) \cdot \Delta A,
\end{equation*}
using the classic calculus approach. Thus \(\iint_R f(x,y) \, dA\) is a limit of double Riemann sums, but while this definition tells us exactly what a double integral is, it is not very helpful for determining the value of a double integral. Fortunately, there is a way to view a double integral as an iterated integral, which will make computations feasible in many cases.
The viewpoint of an iterated integral is closely connected to an important idea from single-variable calculus. When we studied solids of revolution, such as the one shown in Figure 12.2.1, we saw that in some circumstances we could slice the solid perpendicular to an axis and approximate the volume using circular disks. From there, we were able to find the volume of each disk, and then use an integral to add the volumes of the slices to find the total volume of the rotated solid.
We will look at an example of this approach in the following Preview Activity, we will look at how use single variable integrals along traces in a similar way to the rotation volume described above. We will generalize this approach in this section for double integrals over rectangular regions. In Section 12.3, we will look at using this technique on non-rectangular regions of integration.
Preview Activity 12.2.1.
In this activity we will explore the double integral of \(f(x,y) = 25-x^2-y^2\) on the rectangular domain \(R = [-3,3] \times [-4,4]\text{.}\)
As with partial derivatives, we may treat one of the variables in \(f\) as constant and think of the resulting function as a function of a single variable. We will now investigate what this means geometrically and algebraically when we integrate with one input variable fixed.
(a)
Let \(a\) be a fixed value in the interval \([-3,3]\text{.}\) Compute
\begin{equation*}
\int_{-4}^4 f(a,y) dy
\end{equation*}
(b)
The answer of your previous task depended on the value of \(a\) but the process would be the exact same set of steps for every \(a\text{,}\) so we can define a function \(A(x)\) as
\begin{equation*}
A(x) = \int_{-4}^4 f(x,y) \, dy
\end{equation*}
Explain the geometric meaning of the value of \(A(x)\) relative to the surface defined by \(f\) in the context of the trace determined by the fixed value of \(x\text{.}\) Your explanation should refer to different parts of Figure 12.2.2
(c)
For a fixed value of \(x\text{,}\) say \(x_i^*\text{,}\) write a few sentences to explain the geometric meaning of \(A(x_i^*) \ \Delta x\text{.}\) Your explanation of \(A(x_i^*) \Delta x\) should refer to the parts of Figure 12.2.3.
(d)
Since \(f\) is continuous on \(R\text{,}\) we can define the function \(A = A(x)\) at every value of \(x\) in \([-3,3]\text{.}\) Now think about subdividing the \(x\)-interval \([-3,3]\) into \(m\) subintervals, and choosing a value \(x_i^*\) in each of those subintervals. What will be the meaning of the sum \(\sum_{i=1}^m A(x_i^*) \ \Delta x\text{?}\)
(e)
Explain why \(\int_{-3}^3 A(x) \, dx\) will determine the exact value of the volume under the surface \(z = f(x,y)\) over the rectangle \(R\text{.}\)
Subsection 12.2.2 Iterated Integrals
The ideas that we explored in Preview Activity 12.2.1 work more generally and lead to the idea of an iterated integral. Let \(f\) be a continuous function on a rectangular domain \(R = [a,b] \times [c,d]\text{,}\) and let
\begin{equation*}
A(x) = \int_c^d f(x,y) \, dy.
\end{equation*}
The function \(A = A(x)\) determines the value of the cross sectional area (by area we mean “signed” area) in the \(y\) direction for the fixed value of \(x\) of the solid bounded between the surface defined by \(f\) and the \(xy\)-plane.
The value of this cross sectional area is determined by the input \(x\) in \(A\text{.}\) Since \(A\) is a function of \(x\text{,}\) it follows that we can integrate \(A\) with respect to \(x\text{.}\) In doing so, we use a partition of \([a, b]\) and make an approximation to the integral given by
\begin{equation*}
\int_a^b A(x) \, dx \approx \sum_{i=1}^m A(x_i^*) \Delta x,
\end{equation*}
where \(x_i^*\) is any number in the subinterval \([x_{i-1},x_i]\text{.}\) Each term \(A(x_i^*) \Delta x\) in the sum represents an approximation of a fixed cross sectional slice of the surface in the \(y\) direction with a fixed width of \(\Delta x\) as illustrated in Figure 12.2.4. We add the signed volumes of these slices as shown in the frames in Figure 12.2.4 to obtain an approximation of the total signed volume.
As we let the number of subintervals in the \(x\) direction approach infinity, we can see that the Riemann sum \(\sum_{i=1}^m A(x_i^*) \Delta x\) approaches a limit and that limit is the sum of signed volumes bounded by the function \(f\) on \(R\text{.}\) Therefore, since \(A(x)\) is itself determined by an integral, we have
\begin{align*}
\iint_R f(x,y) \, dA \amp= \lim_{m \to \infty} \sum_{i=1}^m A(x_i^*) \Delta x \\
\amp= \int_a^b A(x) \, dx = \int_a^b \left[ \int_c^d f(x,y) \, dy \right] \, dx.
\end{align*}
Hence, we can compute the double integral of \(f\) over \(R\) by first integrating \(f\) with respect to \(y\) on \([c, d]\text{,}\) then integrating the resulting function of \(x\) with respect to \(x\) on \([a, b]\text{.}\) The nested integral
\begin{equation*}
\int_a^b \left[ \int_c^d f(x,y) \, dy \right] \, dx = \int_a^b \int_c^d f(x,y) \, dy \, dx
\end{equation*}
is called an iterated integral, and we see that each double integral may be represented by two single integrals.
We can also think of these iterated integrals as a description for evaluating one of our Riemann sums over fixed values (corresponding to the other input variable). As in our descriptions above, we can partition our space into pieces such that the inner sum of
\begin{equation*}
\sum_{j=1}^m \sum_{i=1}^n f\left(x_{ij}^*, y_{ij}^*\right) \, \Delta A
\end{equation*}
corresponds to our gridding in \(y\) (along a constant value \(x_i^*\)). This corresponds to the inner sum being along a column of smaller rectangles, as shown in Figure 12.2.5.
Our outer sum will then correspond to the sum over each of these cross sectional areas (column sums). Because we are working with a rectangular region of integration, the limit as \(n \to \infty\) will make these column sums converge to integrals of the form \(\int_c^d f(x_i^*,y) \, dy\text{,}\) which we called \(A(x^*_i)\text{.}\) The outer sum will converge to the integral \(\int_a^b A(x) \, dx\text{,}\) giving the iterated integral form.
In our arguments above, we made the choice to integrate first with respect to \(y\) (holding \(x\) constant first). A parallel argument shows that we can also find the double integral as an iterated integral integrating with respect to \(x\) first, or
\begin{equation*}
\iint_R f(x,y) \, dA = \int_c^d \left( \int_a^b f(x,y) \, dx \right) \, dy = \int_c^d \int_a^b f(x,y) \, dx \, dy.
\end{equation*}
You could visualize this as first taking the sum across rows of smaller rectangles in Figure 12.2.5, then taking the sum of this column of sums. This order of operations would correspond to integrating with respect to \(x\) first (holding \(y\) constant for a particular row), then integrating across the range of \(y\) values.
The fact that integrating in either order results in the same value is known as Fubini’s Theorem.
Fubini’s Theorem.
If \(f = f(x,y)\) is a continuous function on a rectangle \(R = [a,b] \times [c,d]\text{,}\) then
\begin{equation*}
\iint_R f(x,y) \, dA = \int_c^d \int_a^b f(x,y) \, dx \, dy = \int_a^b \int_c^d f(x,y) \, dy \, dx.
\end{equation*}
Fubini’s theorem enables us to evaluate double integrals by using iterated integrals instead of resorting to the limit definition. Working with one integral at a time, we can use the Fundamental Theorem of Calculus from single-variable calculus to find the exact value of each integral, starting with the inner integral.
Aside
Activity 12.2.2.
(a)
Viewing \(x\) as a fixed constant, use the Fundamental Theorem of Calculus to evaluate the integral
\begin{equation*}
A(x) = \int_{-4}^4 f(x,y) \, dy.
\end{equation*}
Note that you will be integrating with respect to \(y\text{,}\) and holding \(x\) constant. Your result should be a function of \(x\) only.
(b)
Next, use your result from (a) along with the Fundamental Theorem of Calculus to determine the value of \(\int_{-3}^3 A(x) \, dx\text{.}\)
(c)
What is the value of \(\displaystyle{\iint_R f(x,y) \, dA}\text{?}\) Write a sentence to interpret the meaning of this value for two out of the three different ways mentioned in Subsection 12.1.4.
Activity 12.2.3.
(a)
Evaluate \(\displaystyle{\iint_R f(x,y) \, dA}\) using an iterated integral. Choose an order for integration by deciding whether you want to integrate first with respect to \(x\) or \(y\text{.}\)
(b)
Evaluate \(\displaystyle{\iint_R f(x,y) \, dA}\) using the iterated integral whose order of integration is the opposite of the order you chose in (a).
Subsection 12.2.3 Summary
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We can evaluate the double integral \(\iint_R f(x,y) \, dA\) over a rectangle \(R = [a,b] \times [c,d]\) as an iterated integral in one of two ways:
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\(\displaystyle{\int_a^b \left[ \int_c^d f(x,y) \, dy \right] \, dx}\text{,}\) or
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\(\displaystyle{\int_c^d \left[ \int_a^b f(x,y) \, dx \right] \, dy}\text{.}\)
This process works because each inner integral represents a cross-sectional (signed) area and the outer integral then sums all of the cross-sectional (signed) areas. Fubini’s Theorem guarantees that the resulting value is the same, regardless of the order in which we integrate. -
Exercises 12.2.4 Exercises
1.
Evaluate the iterated integral \(\int_{0}^{4} \int_{0}^{3} 3x^2y^3 \, dx dy\)
2.
Evaluate the iterated integral \(\int_{4}^{5} \int_{2}^{3} (2x + y)^{-2} \: dy dx\)
3.
Find \(\int_{0}^{6} \int_{7}^{8}( x + \ln y) \,dydx\)
4.
Find \(\int_{0}^{4} \int_{0}^{5} xy e^{x+y} \,dydx\)
5.
Calculate the double integral \(\int \int_{\mathbf{R}} (4x + 4y + 16 )\: dA\) where \(\mathbf{R}\) is the region: \(0 \leq x \leq 2, 0 \leq y \leq 2\text{.}\)
6.
Calculate the double integral \(\int \int_{\mathbf{R}} x \cos(x + y) \: dA\) where \(\bf{R}\) is the region: \(0 \leq x \leq \frac{\pi}{3}, 0 \leq y \leq \frac{\pi}{2}\)
7.
Consider the solid that lies above the square (in the xy-plane) \(R = [0, 2] \times [0, 2]\text{,}\)
and below the elliptic paraboloid \(z = 49 - x^{2} - 3y^2\text{.}\)
(A) Estimate the volume by dividing R into 4 equal squares and choosing the sample points to lie in the lower left hand corners.
(B) Estimate the volume by dividing R into 4 equal squares and choosing the sample points to lie in the upper right hand corners..
(C) What is the average of the two answers from (A) and (B)?
(D) Using iterated integrals, compute the exact value of the volume.
8.
If \(\displaystyle \int_{1}^{2} f(x) \: dx = -3\) and \(\displaystyle \int_{3}^{6} g(x) \: dx = -2\text{,}\) what is the value of \(\displaystyle \int\!\!\int_{D} f(x)\!g(y) \: dA\) where \(D\) is the rectangle: \(1 \leq x \leq 2, \ \ 3 \leq y \leq 6\text{?}\)
9.
10.
11.
Evaluate each of the following double or iterated integrals exactly.
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\(\displaystyle \int_1^3 \left( \int_2^5 xy \, dy \right) \, dx\)
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\(\displaystyle \int_0^{\pi/4} \left( \int_0^{\pi/3} \sin(x) \cos(y) \, dx \right) \, dy\)
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\(\displaystyle \int_0^1 \left( \int_0^1 e^{-2x - 3y} \, dy \right) \, dx\)
12.
The temperature at any point on a metal plate in the \(xy\) plane is given by \(T(x,y) = 100-4x^2 - y^2\text{,}\) where \(x\) and \(y\) are measured in inches and \(T\) in degrees Celsius. Consider the portion of the plate that lies on the rectangular region \(R = [1,5] \times [3,6]\text{.}\)
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Write an iterated integral whose value represents the volume under the surface \(T\) over the rectangle \(R\text{.}\)
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Evaluate the iterated integral you determined in (a).
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Find the area of the rectangle, \(R\text{.}\)
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Determine the exact average temperature, \(T_{\operatorname{AVG}(R)}\text{,}\) over the region \(R\text{.}\)
13.
Consider the box with a sloped top that is given by the following description: the base is the rectangle \(R = [1,4] \times [2,5]\text{,}\) while the top is given by the plane \(z = p(x,y) = 30 - x - 2y\text{.}\)
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Write an iterated integral whose value represents the volume under \(p\) over the rectangle \(R\text{.}\)
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Evaluate the iterated integral you determined in (a).
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If you wanted to build a rectangular box (with an identical base) that has the same volume as the box with the sloped top described here, how tall would the rectangular box have to be?
