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13.3 Double Integrals over General Regions In this section, we study how to integrate a function not just over rectangles but also over regions of more general shape. Let D be a bounded region enclosed in a rectangular region R and f (x, y) a continuous function over D. We dene ( F (x, y) = f (x, y) 0 if (x, y) is in D if (x, y) is in R but not in D If F is integrable over R, then we dene the double integral of f over D by ¨ ¨ f (x, y) dA := D F (x, y) dA R : If f (x, y) ≥ 0 and f is continuous on the region D then the volume V of the solid S that lies above D and under the graph of f is Fact ¨ V = f (x, y) dA D Example 1. Evaluate the double integral ˜p D 16 − x2 − y 2 dA by identifying it as a volume of a solid, where D = { (x, y) ∈ R2 | x2 + y 2 ≤ 16 }. : Properties of double integrals 1. If α and β are real numbers, then D D D 2. If f (x, y) ≥ g(x, y), then ¨ ¨ f (x, y) dA ≥ 3. g(x, y) dA D 1 dA = the area of D dA = D D ¨ g(x, y) dA f (x, y) dA + β (αf (x, y) + βg(x, y)) dA = α ¨ ¨ ¨ ¨ D 4. If D = D1 ∪D2 , where D1 and D2 do not overlap except perhaps on their boundaries, then ¨ ¨ ¨ f (x, y) dA = D f (x, y) dA + D1 f (x, y) dA D2 Example 2. If D = { (x, y) ∈ R2 | x2 + y 2 ≤ 16 }, then ˜ D dA = A plane region D is said to be of continuous functions of x, that is, type I if it lies between the graphs of two D = {(x, y) | a ≤ x ≤ b, g1 (x) ≤ y ≤ g2 (x)} where g1 and g2 are continuous on [a, b]. . If D is a region of type I such that D = {(x, y) | a ≤ x ≤ b, g1 (x) ≤ y ≤ g2 (x)}, then Theorem ˆ bˆ ¨ g2 (x) f (x, y) dydx f (x, y) dA = D Example 3. Evaluate the double integral a ˜ D g1 (x) (x + y) dA where D is the region bounded by the lines x = 2, y = x and the hyperbola xy = 1. A plane region D is said to be of continuous functions of y , that is, type II if it lies between the graphs of two D = {(x, y) | c ≤ y ≤ d, h1 (y) ≤ x ≤ h2 (y)} where h1 and h2 are continuous on [c, d]. . If D is a region of type II such that D = {(x, y) | c ≤ y ≤ d, h1 (y) ≤ x ≤ h2 (x)}, then Theorem ¨ ˆ d ˆ h2 (y) f (x, y) dxdy f (x, y) dA = D Example 4. Evaluate the double integral with vertices (0, 0), (2, 4) and (6, 0). h1 (y) c ˜ D xy dA where D is the triangular region Example 5. Find the volume of the solid bounded by the cylinder x2 + y 2 = 1 and the planes x = 0, y = z , z = 0 in the rst octant. Example 6. Evaluate the integral by reversing the order of integration: ˆ 1ˆ 1 x3 sin(y 3 ) dydx 0 x2 Example 7. Sketch the region of integration and change the order of integration: ˆ 1ˆ √ ˆ 2ˆ x f (x, y) dydx f (x, y) dydx + 0 0 2−x 1 0