bài giảng vật lý bằng tiếng anh iteratedintegrals

MULTIPLE INTEGRALS 2.2 Iterated Integrals In this section, we will learn how to: Express double integrals as iterated integrals INTRODUCTION Once we have expressed a double integral as an iterated integral, we can then evaluate it by calculating two single integrals INTRODUCTION Suppose that f is a function of two variables that is integrable on the rectangle R = [a, b] x [c, d] INTRODUCTION We use the notation to mean: ∫ d c f ( x, y ) dy  x is held fixed  f(x, y) is integrated with respect to y from y = c to y = d PARTIAL INTEGRATION This procedure is called partial integration with respect to y Notice its similarity to partial differentiation PARTIAL INTEGRATION d Now, ∫c f ( x, y ) dy is a number that depends on the value of x So, it defines a function of x: d A( x) = ∫ f ( x, y ) dy c Equation PARTIAL INTEGRATION If we now integrate the function A with respect to x from x = a to x = b, we get: ∫ b a A( x) dx = ∫  ∫ f ( x, y ) dy  dx  a   c b d ITERATED INTEGRAL The integral on the right side of Equation is called an iterated integral Equation ITERATED INTEGRALS Thus, b d a c ∫∫ f ( x, y ) dy dx = ∫  ∫ f ( x, y ) dy  dx  a   c b d means that:  First, we integrate with respect to y from c to d  Then, we integrate with respect to x from a to b ITERATED INTEGRALS Similarly, the iterated integral d b c a ∫ ∫ f ( x, y ) dy dx = ∫ d c  b f ( x, y ) dx  dy  ∫a  means that:  First, we integrate with respect to x (holding y fixed) from x = a to x = b  Then, we integrate the resulting function of y with respect to y from y = c to y = d ITERATED INTEGRALS Example Theorem FUBUNI’S THEOREM If f is continuous on the rectangle R = {(x, y) |a ≤ x ≤ b, c ≤ y ≤ d} then b d a c ∫∫ f ( x, y) dA = ∫ ∫ R =∫ d c ∫ b a f ( x, y ) dy dx f ( x, y ) dx dy ITERATED INTEGRALS Example ITERATED INTEGRALS Example ITERATED INTEGRALS To be specific, suppose that: f(x, y) = g(x)h(y) R = [a, b] x [c, d] ITERATED INTEGRALS Then, Fubini’s Theorem gives: d b c a ∫∫ f ( x, y) dA = ∫ ∫ R g ( x)h( y ) dx dy = ∫  ∫ g ( x)h( y ) dx  dy  c   a d b ITERATED INTEGRALS In the inner integral, y is a constant So, h(y) is a constant and we can write: ∫ d c  b g ( x)h( y ) dx  dy = d  h( y ) ∫c   ∫a  (∫ b a b d a c )  g ( x) dx dy  = ∫ g ( x) dx ∫ h( y ) dy since ∫ b a g ( x) dx is a constant Equation ITERATED INTEGRALS Hence, in this case, the double integral of f can be written as the product of two single integrals: b d a c g ( x ) h ( y ) dA = g ( x ) dx h ( y ) dy ∫∫ ∫ ∫ R where R = [a, b] x [c, d] ITERATED INTEGRALS Example