微积分英文课件：chapter15 Multiple Integrals.ppt

1.If f is continuous on a polar region of the form then 2.If f is continuous on a polar region of the form then 3.If f is continuous on a polar region of the form then Solution： Example Find 　　　　　　，　　　　　　　　。　　　　　　　　　　　 D is given by So Example Evaluate　　　　　　　，　　　　　　　　　　　 Solution 2 o We have Example 1 Evaluate ,where D is the region bounded by the parabolas Solution Type I Type II Properties of Double Integral Suppose that functions f and g are continuous on a bounded closed region D. Property 1 The double integral of the sum (or difference) of two functions exists and is equal to the sum (or difference) of their double integrals, that is, Property 2 Property 3 where D is divided into two regions D1 and D2 and the area of D1 ∩ D2 is 0. Property 4 If f(x, y) ≥0 for every (x, y) ∈D, then Property 5 If f(x, y)≤g(x, y) for every (x, y) ∈D, then Moreover, since it follows from Property 5 that hence where S is the area of D. Property 6 Property 7 Suppose that M and m are respectively the maximum and minimum values of function f on D, then where S is the area of D. Property 8 (The Mean Value Theorem for Double Integral) If f(x, y) is continuous on D, then there exists at least a point (ξ,η) in D such that where S is the area of D. f (ξ,η) is called the average Value of f on D Example 2 Evaluate ,where D is the region bounded by the parabolas Solution Type II Type I Example 3 Evaluate ,where D is the region bounded by the parabolas Solution Type I Type II Example change the order of integration solution： We have An alternative description of D is x y o 2 3 1 x y o x y o Example cha