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15.1 Double Integrals over

Rectangles

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Review of the Definite Integral

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Review of the Definite Integral

First let’s recall the basic facts concerning definite integrals of functions of a single variable.

If f(x) is defined for a ≤ x ≤ b, we start by dividing the interval [a, b] into n subintervals [xi–1, xi] of equal width

∆x = (b – a)/n and we choose sample points in these subintervals. Then we form the Riemann sum

and take the limit of such sums as n → to obtain the definite integral of f from a to b:

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Review of the Definite Integral

In the special case where f(x) ≥ 0, the Riemann sum can be interpreted as the sum of the areas of the approximating

rectangles in Figure 1, and represents the area under the curve y = f(x) from a to b.

Figure 1

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Volumes and Double Integrals

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In a similar manner we consider a function f of two variables defined on a closed rectangle

R = [a, b] × [c, d] = {(x, y) ∈ |a ≤ x ≤ b, c ≤ y ≤ d}

and we first suppose that f(x, y) ≥ 0.

The graph of f is a surface with equation z = f(x, y).

Let S be the solid that lies above R and under the graph of f, that is,

S = {(x, y, z) ∈ |0 ≤ z ≤ f(x, y), (x, y) ∈ R}

(See Figure 2.)

Volumes and Double Integrals

Figure 2

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Volumes and Double Integrals

Our goal is to find the volume of S.

The first step is to divide the rectangle R into subrectangles.

We accomplish this by dividing the interval [a, b] into m subintervals [xi–1, xi] of equal width ∆x = (b – a)/m and dividing [c, d ] into n subintervals [yj– 1, yj] of equal width

∆y = (d – c)/n.

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Volumes and Double Integrals

By drawing lines parallel to the coordinate axes through the endpoints of these subintervals, as in Figure 3, we form the subrectangles

Rij = [xi– 1, xi] × [yj–1, yj] = {(x, y) | xi– 1 ≤ x ≤ xi, yj–1 ≤ y ≤ yj} each with area ∆A = ∆x ∆y.

Figure 3

Dividing R into subrectangles

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Volumes and Double Integrals

If we choose a sample point in each Rij, then we can approximate the part of S that lies above each Rij by a

thin rectangular box (or “column”) with base Rij and height as shown in Figure 4.

The volume of this box is the height of the box times the area of the base rectangle:

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Volumes and Double Integrals

If we follow this procedure for all the rectangles and add the volumes of the corresponding boxes, we get an

approximation to the total volume of S:

(See Figure 5.) This double sum means that for each subrectangle we evaluate f at the chosen point and multiply by the area of the subrectangle, and then we

add the results. Figure 5

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Volumes and Double Integrals

Our intuition tells us that the approximation given in (3) becomes better as m and n become larger and so we would expect that

We use the expression in Equation 4 to define the volume of the solid S that lies under the graph of f and above the rectangle R.

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Volumes and Double Integrals

Limits of the type that appear in Equation 4 occur

frequently, not just in finding volumes but in a variety of other situations even when f is not a positive function. So we make the following definition.

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Volumes and Double Integrals

The precise meaning of the limit in Definition 5 is that for every number ε > 0 there is an integer N such that

for all integers m and n greater than N and for any choice of sample points in Rij.

A function f is called integrable if the limit in Definition 5 exists.

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Volumes and Double Integrals

It is shown in courses on advanced calculus that all continuous functions are integrable. In fact, the double

integral of f exists provided that f is “not too discontinuous.”

In particular, if f is bounded on R, [that is, there is a

constant M such that | f(x, y)| ≤ M for all (x, y) in R], and f is continuous there, except on a finite number of smooth

curves, then f is integrable over R.

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Volumes and Double Integrals

The sample point can be chosen to be any point in the subrectangle Rij, but if we choose it to be the upper right-hand corner of Rij [namely (xi, yj), see Figure 3], then the expression for the double integral looks simpler:

Figure 3

Dividing R into subrectangles

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Volumes and Double Integrals

By comparing Definitions 4 and 5, we see that a volume can be written as a double integral:

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Volumes and Double Integrals

The sum in Definition 5,

is called a double Riemann sum and is used as an

approximation to the value of the double integral. [Notice how similar it is to the Riemann sum in (1) for a function of a single variable.]

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Volumes and Double Integrals

If f happens to be a positive function, then the double

Riemann sum represents the sum of volumes of columns, as in Figure 5, and is an approximation to the volume under the graph of f.

Figure 5

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Example 1

Estimate the volume of the solid that lies above the square R = [0, 2] × [0, 2] and below the elliptic paraboloid

z = 16 – x2 – 2y2. Divide R into four equal squares and choose the sample point to be the upper right corner of each square Rij. Sketch the solid and the approximating rectangular boxes.

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Example 1 – Solution

The squares are shown in Figure 6.

The paraboloid is the graph of f(x, y) = 16 – x2 – 2y2 and the area of each square is ∆A = 1.

Figure 6

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Example 1 – Solution

Approximating the volume by the Riemann sum with m = n = 2, we have

= f(1, 1) ∆A + f(1, 2) ∆A + f(2, 1) ∆A + f(2, 2) ∆A

= 13(1) + 7(1) + 10(1) + 4(1)

= 34

cont’d

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Example 1 – Solution

This is the volume of the approximating rectangular boxes shown in Figure 7.

Figure 7

cont’d

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The Midpoint Rule

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The Midpoint Rule

The methods that we used for approximating single integrals (the Midpoint Rule, the Trapezoidal Rule,

Simpson’s Rule) all have counterparts for double integrals.

Here we consider only the Midpoint Rule for double integrals.

This means that we use a double Riemann sum to

approximate the double integral, where the sample point in Rij is chosen to be the center of Rij. In

other words, is the midpoint of [xi–1, xi] and is the midpoint of [yj– 1, yj].

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The Midpoint Rule

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Example 3

Use the Midpoint Rule with m = n = 2 to estimate the value of the integral ∫∫R(x – 3y2) dA, where

R = {(x, y) | 0 ≤ x ≤ 2, 1 ≤ y ≤ 2}.

Solution:

In using the Midpoint Rule with m = n = 2, we evaluate f(x, y) = x – 3y2 at the centers of the four subrectangles shown in Figure 10.

Figure 10

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Example 3 – Solution

So and The area of each subrectangle is Thus

cont’d

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Example 3 – Solution

Thus we have

cont’d

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Iterated Integrals

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Iterated Integrals

Suppose that f is a function of two variables that is integrable on the rectangle R = [a, b] × [c, d].

We use the notation to mean that x is held fixed and f(x, y) is integrated with respect to y from y = c to y = d.

This procedure is called partial integration with respect to y.

(Notice its similarity to partial differentiation.)

Now is a number that depends on the value of x, so it defines a function of x:

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Iterated Integrals

If we now integrate the function A with respect to x from x = a to x = b, we get

The integral on the right side of Equation 7 is called an iterated integral. Usually the brackets are omitted. Thus

means that we first integrate with respect to y from c to d

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Iterated Integrals

Similarly, the iterated integral

means that we first integrate with respect to x (holding y fixed) from x = a to x = b and then we integrate the

resulting function of y with respect to y from y = c to y = d.

Notice that in both Equations 8 and 9 we work from the inside out.

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Example 4

Evaluate the iterated integrals.

(a) (b)

Solution:

(a) Regarding x as a constant, we obtain

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Example 4 – Solution

Thus the function A in the preceding discussion is given by in this example.

We now integrate this function of x from 0 to 3:

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Example 4 – Solution

(b) Here we first integrate with respect to x:

cont’d

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Iterated Integrals

Notice that in Example 4 we obtained the same answer whether we integrated with respect to y or x first.

In general, it turns out (see Theorem 10) that the two

iterated integrals in Equations 8 and 9 are always equal;

that is, the order of integration does not matter. (This is similar to Clairaut’s Theorem on the equality of the mixed partial derivatives.)

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Iterated Integrals

The following theorem gives a practical method for

evaluating a double integral by expressing it as an iterated integral (in either order).

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Iterated Integrals

In the special case where f(x, y) can be factored as the product of a function of x only and a function of y only, the double integral of f can be written in a particularly simple form.

To be specific, suppose that f(x, y) = g(x)h(y) and R = [a, b] × [c, d].

Then Fubini’s Theorem gives

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Iterated Integrals

In the inner integral, y is a constant, so h(y) is a constant and we can write

since is a constant.

Therefore, in this case, the double integral of f can be written as the product of two single integrals:

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Average Value

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Average Value

We know that the average value of a function f of one variable defined on an interval [a, b] is

In a similar fashion we define the average value of a function f of two variables defined on a rectangle R to be

where A(R) is the area of R.

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Average Value

If f(x, y) ≥ 0, the equation

A(R) × fave = f(x, y) dA

says that the box with base R and height fave has the same volume as the solid that lies under the graph of f.

[If z = f(x, y) describes a

mountainous region and you chop off the tops of the

mountains at height fave, then you can use them to fill in the valleys so that the region

becomes completely flat.

See Figure 17.] Figure 17

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Example 9

The contour map in Figure 18 shows the snowfall, in inches, that fell on the state of Colorado on December 20 and 21, 2006. (The state is in the shape of a rectangle that

measures 388 mi west to east and 276 mi south to north.) Use the contour map to estimate the average snowfall for the entire state of Colorado on those days.

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Example 9 – Solution

Let’s place the origin at the southwest corner of the state.

Then 0 ≤ x ≤ 388, 0 ≤ y ≤ 276, and f(x, y) is the snowfall, in inches, at a location x miles to the east and y miles to the north of the origin.

If R is the rectangle that represents Colorado, then the average snowfall for the state on December 20–21 was

where A(R) = 388 276.

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Example 9 – Solution

To estimate the value of this double integral, let’s use the Midpoint Rule with m = n = 4. In other words, we divide R into 16 subrectangles of equal size, as in Figure 19.

cont’d

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Example 9 – Solution

The area of each subrectangle is

Using the contour map to estimate the value of f at the center of each subrectangle, we get

≈ ∆A[0 + 15 + 8 + 7 + 2 + 25 + 18.5 + 11 + 4.5 + 28 + 17 + 13.5 + 12 + 15 + 17.5 + 13]

= (6693)(207)

= 6693 mi2

cont’d

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Example 9 – Solution

Therefore

On December 20–21, 2006, Colorado received an average of approximately 13 inches of snow.

cont’d

Figure

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