《微积分教学资料》cha

1、Chapter 5 Applications of integrals Areas between curves The area A of the region bounded by the curves And the lines , where and are continuous and for all in is )(),(xgyxfybxax , fg)()(xgxf x , ba dxxgxfA b a )()( )(xfy )(xgy a b x y A The Riemann sum xxgxf i n i i )()( 1 dxxgxf xxgxfA b a i n i i

2、 n )()( )()(lim 1 Therefore Example 1 Find the area of the region bounded Above by bounded below by and bounded on the sides by x ey xy 1and0 xx Solution 0 x y yx x ye 1 0 () x Aex dx 21 0 1 () 2 x ex 1.5e The area A between the curves And between is ( )( )andyf xyg x andxaxb ( )( ) ( ( )( )( ( )( )

3、 b a cb ac Af xg x dx g xf x dxf xg x dx ( )yf x ( )yg x x y 0 a b c Example 2 Find the area of the region bounded by the curves sin ,cos ,0. 2 andyx yx xx Solution2 0 sincosAxx dx sinyx cosyx 0 4 2 4 0 2 4 (cossin ) (sincos ) xx dx xx dx 42 0 4 sincos cossin xxxx 2 22 The area A of the region bound

4、ed by the curves And the lines , where and are continuous and for all in is ( ),( )xy xy,yc yd ( )( )yy y , cd ( )( ) d c Ayy dy )( yx x y c d )(yx o 2 2 xy xy We find that the points of intersection are(0, 0) and (1, 1).So x o y xy 2 2 xy ) 1 , 1 ( 1 3 1 1 2 0 ()Axxdx 1 0 33 2 3 2 3 x x 3 1 1 2 0 (

5、)Ayydy 1 0 33 2 3 2 3 y y Example 3 Find the area enclosed by the curves 22 andyxyx Solution By solving the system of equations or Example 3 Find the area enclosed by the curves 2 24andby the lineyxyx 2 2 4 yx yx We find that the points of intersection are(2, -2) and (8, 4).So Solution By solving th

6、e system of equations 4 2 2 1 (4) 2 Ayydy 18 2 4 ) 6 4 2 ( 32 y y y )4 , 8( )2, 2( x o y xy2 2 4xy )4 , 8( )2, 2( x o y xy2 2 4xy -2 42 0 22Axdx 8 2 )4(2dxxx 18 or Volumes Definition of volume Let s be a solid that lies between and.xaxbIf the cross-sectional area of s in the plane x pThrough and per

7、pendicular to the x axis,is A( ) ,where is a continuous function,xxA then the volume of s is * 1 lim()( ) n b i an i VA xxA x dx x oa b A(x) x ,bxxxxxxa nii 1210 )., 2, 1( n -b 1 ni a xxx ii n i i VV 1 , 1 * iii xxx ), 2 , 1(ni i VxxA i )( * i n i i xxA )( 1 * 分点为：分点为： x a b A( ) 1i x i x * i x * i

8、x o 1) Partition: 2) Approximation: 3) Sum: 4) Limit: V xxA n i i n )(lim 1 * b a dxxA)( 22 1 ( )1616tan30 2 A xxx 4 2 4 1 (16) 2 3 Vxdx 3 4 11 16 432 3 xx x 30 o x y 4 4 22 16xy 30 128 3 3 Example A wedge is cut out of a circular cylinder of radius 4 by two planes.0ne plane is perpendicular to the

9、axis of the cylinder.the other intersects the first at an angle of along a diameter of the cylinder.find the volume of the wedge. 30 Solution The cross-sectional area is the volume of a solid of revolution 、 Solids of revolution: x 2 2 ( ) ( ) b a b a Vf xdx f xdx )( xfy ab x y o The volume of the s

10、olid obtained by rotating the region bounded by ),(xfy , ax and,xb 0y about the xaxis The cross-sectional area is 22 ( ) ( )A xradiusf x 22 ( ( )( ( ) b a Vf xg xdx 2.The volume of the solid obtained by rotating the region bounded by ),(xfy , ax and,xb ( ),( )( )yg xf xg x about the xaxis The cross-

11、sectional area is 22 22 ( ) ( ) ( ) A xout radiusinner radius f xg x )(xfy )(xgy a b y A x y c d o )(yx dy)y(V d c 2 3. Example Find the volume of the solid obtained by rotating about the x-axis the region under the curve from 0 to 1. yx Solution 1 0 1 0 ( )VA x dx xdx 2 1 0 22 x Example Find the vo

12、lume of the solid obtained by rotating about the y axis the region bounded by 3, 8and0.yxyx Solution 2 22 33 ( )()A yxyy The cross-sectional area is 2 88 3 00 96 ( ) 5 VA y dyy dy The solid lies between y=0 and y=8 ,its volume is 2 22 33 ( )()A yxyy The cross-sectional area is The solid lies between

13、 y=0 and y=8 ,its volume is 2 22 33 ( )()A yxyy The cross-sectional area is y x o 31 2 -2 2 22 21 2 ( )( )Vxyxy dy 2 0 2 1 2 2 )()(2dyyxyx 2 2 43yx 2 1 43yx 4) 3( 22 yx Example Find the volume of the solid obtained by rotating the region bounded by about the y-axis Solution 22 12 ( ) ( )( )A yx yx y

14、 2 24 2 0 2222 )43()43(2dyyy 2 0 2 424dyy Example The region enclosed by the curves is rotated about the x-axis. Find the volume of the resulting solid. 2 yx and yx Solution 2 yx yx (0,0) (1,1) 11 24 00 2 ( )() 15 VA x dxxxdx The solid lies between x=0 and x=1 ,its volume is 22224 ( )( )()()A xxxxx The cross-sectional area is Example The region enclosed by the curves the