台湾成功大学课件 Double Integrals_微积分（二）

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Double Integrals The Double Integral over a Rectangle We start with a function fcontinuous on a rectangle R:a≤x≤b,c≤y≤d We want to define the double integral of fover R: 「f(x,y)d. d R a b Salas,Hille,Etgen Calculus:One and Several Varisbles Figure 17.2.1 Copy right 2007 John Wiley Sors,Inc.All rights reserved

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Double Integrals The Double Integral over a Rectangle We start with a function f continuous on a rectangle R : a ≤ x ≤ b, c ≤ y ≤ d We want to define the double integral of f over R: ( , . ) R f x y dxdy 

Let P={xox1,...,xm}be a partition of [a,b] and P2={yoy1,..·,yn} a partition of [c,dl. Then the set P=P1XP2={x,y):x∈P,y∈P2} is called a partition of R (xy d=yn 方 y1 c=yo a=x0尤1 Xi xm=b X Figure 17.2.2 d Several Varisbles ne.All rights reserved

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Let P1 = {x0 , x1 , . . . , xm} be a partition of [a, b] and P2 = {y0 , y1 , . . . , yn} a partition of [c, d] . Then the set P = P1 × P2 = {(xi , yj ) : xi  P1 , yj  P2} is called a partition of R

Double Integrals The sum of all the products M.,(area of R)=M (x,-x)(v,-y)=M.Ax,Av, is called the P upper sum forf: 71 (17.2.1) U(P)=∑∑Mi(aea ofR)=∑∑M,AAy. i=1j=1 i=1j=1 Salas,Hille,Etgen Calculus:One and Several Varisbles Main Menu Copy right 2007 John Wiley Sors,Inc.All rights reserved

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Double Integrals The sum of all the products is called the P upper sum for f : M R M x x y y M x y i j i j i j i i j j i j i j (area of ) = − − =   ( − − 1 1 )( )

The sum of all the products marea of R)=m(x)(yy)=mAxAy is called the P lower sum for f: 17m n m n (17.2.2) Lr(P)-∑∑m((area of Ri)=∑A mi△xi△yi i=1j=1 i=1i=1 Sals,Hille,Etgen Calculus One and Several Main Menu Copy right 2007 John Wiley Sors,Inc.All rights reserved

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The sum of all the products is called the P lower sum for f : m R m x x y y m x y i j i j i j i i j j i j i j (area of ) = − − =   ( − − 1 1 )( )

Double Integrals DEFINITION 17.2.3 THE DOUBLE INTEGRAL OVER A RECTANGLE R Let f be continuous on a closed rectangle R.The unique number I that satisfies the inequality Lr(P)≤I≤U(P) for all partitions P of R is called the double integral of f over R,and is denoted by r. R Salas,Hille,Etgen Calculus:One and Several Varisbles Main Menu Copy right 2007 John Wiley Sors,Inc.All rights reserved

Double Integrals The Double Integral as a Volume If fis continuous and nonnegative on the rectangle R,the equation z=f(x,y) represents a surface that lies above R.In this case the double integral ∬f(x) gives the volume of the solid that is bounded below by R and bounded above by the surface z=f(x,y :=f(r,y)- 子R Figure 17.2.6 Figure 17.2.5 Salas,Hille,Etgen Calculus:One and Several Varisbles Main Menu Copy right 2007 John Wiley Sors,Inc.All rights reserved

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Double Integrals The Double Integral as a Volume If f is continuous and nonnegative on the rectangle R, the equation z = f (x, y) represents a surface that lies above R. In this case the double integral gives the volume of the solid that is bounded below by R and bounded above by the surface z = f (x, y). ( ) R f x dxdy 

Double Integrals Since the choice of a partition P is arbitrary,the volume of Tmust be the double integral: (17.2.4) f(x,y)dx dy. The double integral ∬1k少=∬k少 R R gives the volume of a solid of constant height 1 erected over R.In square units this is just the area of R: (17.2.5) area of R= dx dy. Sals,Hille,Etgen Calculus One and Several Main Menu Copy right 2007 John Wiley Sors,Inc.All rights reserved

Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Double Integrals Since the choice of a partition P is arbitrary, the volume of T must be the double integral: The double integral gives the volume of a solid of constant height 1 erected over R. In square units this is just the area of R: 1 R R dxdy dxdy =  

Example 2.Evaluate adxdy R where a is a constant and R is the rectangle R:a≤x≤b,c≤y≤d. Salas,Hille,Etgen Calculus:One and Several Varisbles Main Menu Copy right 2007 John Wiley Sors,Inc.All rights reserved

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台湾成功大学课件 Double Integrals_微积分（二）

台湾成功大学 课件 Double Integrals_微积分（二）

台湾成功大学课件 Double Integrals_微积分（二）