The Evaluation of Double Integrals By Repeated Integrals Type I Region The projection of onto the x-axis is a closed interval [a,b]and consists of all points(xy)with a≤x≤band (x)≤y≤4(x) y=02(x) 2 y=(x) TypeI p2(x)】 (17.3.1) f(x.y)dx dy=
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Evaluation of Double Integrals By Repeated Integrals Type I Region The projection of Ω onto the x-axis is a closed interval [a, b] and Ω consists of all points (x, y) with a ≤ x ≤ b and 1 2 ( x y x ) ( )
The Evaluation of Double Integrals By Repeated Integrals Type II Region The projection of onto the y-axis is a closed interval [c,d]and consists of all points (x,y)with c≤y≤d and y1y)≤x≤20y), x=(y) x=4(y) Type II (17.3.2) f(x,y)dx dy= fx,y))d
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Evaluation of Double Integrals By Repeated Integrals Type II Region The projection of Ω onto the y-axis is a closed interval [c, d] and Ω consists of all points (x, y) with c ≤ y ≤ d and ψ1 (y) ≤ x ≤ ψ2 (y)
The Evaluation of Double Integrals By Repeated Integrals The Reduction Formulas Viewed Geometrically Suppose that fis nonnegative and is a region of Type I. j∬f(xy)a床=volume of 0 :=f(x,y) volume by double integration Figure 17.3.2 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Evaluation of Double Integrals By Repeated Integrals The Reduction Formulas Viewed Geometrically Suppose that f is nonnegative and Ω is a region of Type I
z=f(x,y) area =M(x) volume by parallel cross sections b p2(x) (2) 「f(xy)y dx=volume of T (x) Main Menu C
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. ( ) ( ) ( ) 2 ( ) 1 2 , volume of x b a x f x y dy dx T =
Example 1 Evaluate -2y)dxdy with as in Figure 16.3.4. Figure 16.3.4 Main Menu c墙8的
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved
Example 2 Evaluate (xy-y3)dxdy with S as in Figure 16.3.5 y=1- =0 x=-1 x=1 Figure 16.3.5 Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved
Example 3 Evaluate (x)dxdy with as in Figure 16.3.6. y=x21(x=v1/2) (1,1) y=x1/4 (x=y4 (0,0) 1 Figure 16.3.6 Main Menu C 007e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved
Example 4 Use double integration to calculate the area of the region enclosed by y=x2 and x+y=2. y=2-x (-2,4 2 (1.1) -2 -1 Figure 16.3.7 Main Menu o
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved
、x=2-y (-2,4) 4 = x=-vy (1,1) Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved
The Evaluation of Double Integrals By Repeated Integrals Symmetry in Double Integration Suppose that is symmetric about the y-axis Iff is odd in x [f(-xy)=-f(x.y)].then ∬f(x,y)k少=0 Iff is even in x [f(-x.y)=f(x.y)].then ∬f(x,y)=2∬f(x,y)d Suppose that is symmetric about the x-axis. Iff is odd in y [f(x,-y)=-f(x,y)].then [f(x,y)=0 Iff is even in y[f(x-y)=f(x y)],then jf(x,)=2∬fx,y)k Main Menu C 007e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Evaluation of Double Integrals By Repeated Integrals Suppose that Ω is symmetric about the y-axis. If f is odd in x [ f (−x, y) = −f (x, y)], then If f is even in x [ f (−x, y) = f (x, y)], then Suppose that Ω is symmetric about the x-axis. If f is odd in y [ f (x,−y) = −f (x, y)], then If f is even in y [ f (x,−y) = f (x, y)], then Symmetry in Double Integration f x y dxdy ( , 0 ) = ( ) ( ) right half of f x y dxdy f x y dxdy , 2 , = f x y dxdy ( , 0 ) = ( ) ( ) upper half of f x y dxdy f x y dxdy , 2 , =