微积分教学资料chapter5

1、chapter 5 applications of integralsareas 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)()(xgxfx, badxxgxfaba )()()(xfy)(xgyabxyathe riemann sumxxgxfinii)()(1dxxgxfxxgxfabainiin )()()()(lim1thereforeexample

2、 1 find the area of the region boundedabove by bounded below by and bounded on the sides by xey xy 1and0 xxsolution0 xyyxxye10()xaex dx2101()2xex1.5e the area a between the curves and between is ( )( )andyf xyg xandxaxb( )( )( ( )( )( ( )( )bacbacaf xg x dxg xf x dxf xg x dx( )yf x( )yg xxy0abcexamp

3、le 2 find the area of the region bounded by the curvessin ,cos ,0.2andyx yx xxsolution20sincosaxx dxsinyxcosyx0424024(cossin )(sincos )xx dxxx dx4204sincos cossin xxxx 2 22 the area a of the region bounded by the curves and the lines , where and are continuous and for all in is ( ),( )xy xy,yc yd( )

4、( )yyy , cd ( )( )dcayy dy)( yxxycd)(yxo22xyxywe find that the points of intersection are(0, 0) and (1, 1).soxoyxy 22xy ) 1 , 1 (131120()axxdx10332323xx 31120()ayydy10332323yy example 3 find the area enclosed by the curves22andyxyxsolutionby solving the system of equationsorexample 3 find the area e

5、nclosed by the curves224and by the lineyxyx224yxyxwe find that the points of intersection are(2, -2) and (8, 4).sosolutionby solving the system of equations4221(4)2ayydy1824)642(32yyy)4 , 8()2, 2( xoyxy224xy)4 , 8()2, 2( xoyxy224xy-242022axdx82)4(2dxxx18orvolumesdefinition of volume let s be a solid

6、 that lies between and.xaxbif the cross-sectional area of s inthe plane xpthrough and perpendicular to the xaxis,is a( ) ,where is a continuous function,xxathen the volume of s is *1lim()( )nbianiva xxa x dx xoaba(x)x,bxxxxxxanii 1210)., 2, 1( n-b 1niaxxxiiniivv1,1*iiixxx), 2 , 1(niivxxai)(*iniixxa)

7、(1*分点为：分点为：xaba( )1ixix*ix*ixo1) partition:2) approximation:3) sum:4) limit:v xxaniin)(lim1*badxxa)(221( )1616tan302a xxx4241(16) 2 3vxdx341116432 3xxx30oxy442216xy301283 3example a wedge is cut out of a circular cylinder of radius 4 by two planes.0ne plane is perpendicular tothe axis of the cylinde

8、r.the other intersects the first at an angle of along a diameter of the cylinder.find the volume of the wedge. 30solutionthe cross-sectional area isthe volume of a solid of revolution 、solids of revolution:x22 ( ) ( )babavf xdxf xdx)( xfy abxyo1. the volume of the solid obtained by rotatingthe regio

9、n bounded by ),(xfy , ax and,xb0y about the xaxisthe cross-sectional area is22( ) ( )a xradiusf x22( ( )( ( ) bavf xg xdx2.the volume of the solid obtained by rotating the region bounded by ),(xfy , ax and,xb( ),( )( )yg xf xg xabout the xaxisthe cross-sectional area is2222( ) ( ) ( )a xout radiusin

10、ner radiusf xg x)(xfy)(xgyabyaxycdo)(yxdy)y(vdc2 3.example find the volume of the solid obtained byrotating about the x-axis the region under the curve from 0 to 1. yxsolution1010( )va x dxxdx21022xexample find the volume of the solid obtained byrotating about the y axis the region bounded by 3,8and

11、0.yxyxsolution22233( )()a yxyythe cross-sectional area is28830096( )5va y dyy dythe solid lies between y=0 andy=8 ,its volume is22233( )()a yxyythe cross-sectional area isthe solid lies between y=0 andy=8 ,its volume is22233( )()a yxyythe cross-sectional area isyxo312-2222212( )( )vxyxy dy202122)()(

12、2dyyxyx2243yx2143yx4) 3(22yxexample find the volume of the solid obtained byrotating the region bounded by about the y-axis solution2212( ) ( )( )a yx yx y224202222)43()43(2dyyy202424dyyexample the region enclosed by the curves is rotated about the x-axis. findthe volume of the resulting solid. 2yx and yxsolution2yxyx(0,0)(1,1)1124002( )()15va x dxxxdxthe solid lies between x=0 andx=1 ,its volume is22224( )( )()()a xxxxxthe cross-sectional area isexample the region enclosed by the curves is rotated about the y=2.findt