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Copyright © Cengage Learning. All rights reserved.

15 Multiple Integrals Copyright © Cengage Learning. All rights reserved.

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15.3 Double Integrals over General Regions

Copyright © Cengage Learning. All rights reserved.

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Double Integrals over General Regions

For single integrals, the region over which we integrate is always an interval. But for double integrals, we want to be able to integrate a function f not just over rectangles but also over regions D of more general shape, such as the one illustrated in Figure 1. Figure 1

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Double Integrals over General Regions

We suppose that D is a bounded region, which means that D can be enclosed in a rectangular region R as in Figure 2. Then we define a new function F with domain R by Figure 2

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Double Integrals over General Regions

If F is integrable over R, then we define the double integral of f over D by Definition 2 makes sense because R is a rectangle and so R F (x, y) dA has been previously defined.

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Double Integrals over General Regions

The procedure that we have used is reasonable because the values of F (x, y) are 0 when (x, y) lies outside D and so they contribute nothing to the integral. This means that it doesn’t matter what rectangle R we use as long as it contains D. In the case where f (x, y) 0, we can still interpret D f (x, y) dA as the volume of the solid that lies above D and under the surface z = f (x, y) (the graph of f).

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Double Integrals over General Regions

You can see that this is reasonable by comparing the graphs of f and F in Figures 3 and 4 and remembering that R F (x, y) dA is the volume under the graph of F. Figure 3 Figure 4

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Double Integrals over General Regions

Figure 4 also shows that F is likely to have discontinuities at the boundary points of D. Nonetheless, if f is continuous on D and the boundary curve of D is “well behaved”, then it can be shown that R F (x, y) dA exists and therefore D f (x, y) dA exists. In particular, this is the case for type I and type II regions.

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Double Integrals over General Regions

A plane region D is said to be of type I if it lies between the graphs of two continuous functions of x, that is, D = {(x, y) | a x b, g1 (x) y g2 (x)} where g1 and g2 are continuous on [a, b]. Some examples of type I regions are shown in Figure 5. Figure 5 Some type I regions

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Double Integrals over General Regions

In order to evaluate D f (x, y) dA when D is a region of type I, we choose a rectangle R = [a, b] [c, d ] that contains D, as in Figure 6, and we let F be the function given by Equation 1; that is, F agrees with f on D and F is 0 outside D. Figure 6

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Double Integrals over General Regions

Then, by Fubini’s Theorem, Observe that F (x, y) = 0 if y < g1 (x) or y > g2 (x) because (x, y) then lies outside D. Therefore because F (x, y) = f (x, y) when g1 (x) y g2 (x).

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Double Integrals over General Regions

Thus we have the following formula that enables us to evaluate the double integral as an iterated integral. The integral on the right side of is an iterated integral, except that in the inner integral we regard x as being constant not only in f (x, y) but also in the limits of integration, g1 (x) and g2 (x).

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Double Integrals over General Regions

We also consider plane regions of type II, which can be expressed as D = {(x, y) | c y d, h1(y) x h2(y)} where h1 and h2 are continuous. Two such regions are illustrated in Figure 7. Figure 7 Some type II regions

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Double Integrals over General Regions

Using the same methods that were used in establishing , we can show that

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Example 1 Evaluate D (x + 2y) dA, where D is the region bounded by the parabolas y = 2×2 and y = 1 + x2. Solution: The parabolas intersect when 2×2 = 1 + x2, that is, x2 = 1, so x = 1. We note that the region D, sketched in Figure 8, is a type I region but not a type II region and we can write D = {(x, y) | –1 x 1, 2×2 y 1 + x2} Figure 8

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Example 1 – Solution cont’d Since the lower boundary is y = 2×2 and the upper boundary is y = 1 + x2, Equation 3 gives

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Example 1 – Solution cont’d

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Properties of Double Integrals

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Properties of Double Integrals

We assume that all of the following integrals exist. The first three properties of double integrals over a region D follow immediately from Definition 2. If f (x, y) g (x, y) for all (x, y) in D, then

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Properties of Double Integrals

The next property of double integrals is similar to the property of single integrals given by the equation If D = D1 U D2, where D1 and D2 don’t overlap except perhaps on their boundaries (see Figure 17), then Figure 17

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Properties of Double Integrals

Property 9 can be used to evaluate double integrals over regions D that are neither type I nor type II but can be expressed as a union of regions of type I or type II Figure 18 illustrates this procedure. D is neither type I nor type II. D = D1 D2, D1 is type I, D2 is type II. Figure 18(a) Figure 18(b)

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Properties of Double Integrals

The next property of integrals says that if we integrate the constant function f (x, y) = 1 over a region D, we get the area of D:

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Properties of Double Integrals

Figure 19 illustrates why Equation 10 is true: A solid cylinder whose base is D and whose height is 1 has volume A(D) 1 = A(D), but we know that we can also write its volume as D 1 dA. Figure 19 Cylinder with base D and height 1

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Properties of Double Integrals

Finally, we can combine Properties 7, 8, and 10 to prove the following property.

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Example 6 Use Property 11 to estimate the integral D esin x cos y dA, where D is the disk with center the origin and radius 2. Solution: Since –1 sin x 1 and –1 cos y 1, we have –1 sin x cos y 1 and therefore e–1 esin x cos y e1 = e Thus, using m = e–1 = 1/e, M = e, and A(D) = (2)2 in Property 11, we obtain

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