6.2-定积分的几何应用-(2)体积

December2004二、体积VolumesDecember20041.已知平行截面面积求立体的体积设有位于区间[a,b]上的一立体abDecember2004ab[,]xab已知立体的垂直于x轴的截面的面积为A(x)x()Ax求立体的体积VDecember2004ab用微元法求体积微元dVx()Axxdx()AxdxV()AxdxdV()dVAxdxDecember2004abx()Axxdx()dVAxdx立体的体积:baVdV()baAxdx()baVAxdx切片法SectionmethodDecember2004例9aaOxy22yaxx22ax22tanax22221()tan2AxaxaxDecember2004aaOxy22yaxx22ax22tanax22221()tan2Axaxax221()tan()2axaxa()aaVAxdx221()tan2aaaxdx32tan3aDecember2004aaOxy22yax另解自学December20042.旋转体的体积()yfxabAba()yfx由y=f(x),y=0,x=a,x=b所围成的图形绕x轴旋转一周，得一旋转体求旋转体的体积VxDecember2004ba()yfx用切片法xx处的切片为一圆片半径：y=f(x)22()[()]Axyfx旋转体的体积()bxaVAxdx2baydx2[()]bafxdxDecember2004ba()yfxx旋转体的体积2[()]bxaVfxdx体积微元为一圆片：2[()]dVfxdx2()[()]Axfxdx故此法称为圆片法Diskmethod《高等数学学习手册》165页December2004由x=f(y),x=0,y=c,y=d所围成的图形绕y轴旋转一周，得一旋转体求旋转体的体积Vycd()xfycd()xfyADecember2004cd()xfy旋转体的体积2[()]dycVfydy体积微元为一圆片：2[()]dVfydyydy2()[()]AyfydyDecember2004由y=f(x),y=g(x),x=a,x=b所围成的图形绕x轴旋转一周，得一旋转体求旋转体的体积Vx()()0()fxgxaxbbaab()fx()gxADecember2004旋转体的体积Vxba()fx()gxVVV大小－2[]()badxxf＝2[]()badxxg－22{[][]()}()bagxdfxx＝December200422{[()][()]}bxaVfxgxdx＝22{[][]())}(dVgdxfxx＝体积微元为：故此法称为垫圈法Washermethod形如一垫圈ba()fx()gx《高等数学学习手册》166页December2004垫圈法WasherMethod例7求椭圆所围成的平面图形分别绕x轴和y轴旋转一周而成的旋转体的体积.22221xyabab解1ADecember200422221xyabab解只需求第一象限部分绕x轴旋转一周而成的旋转体的体积，再乘以2.1A12xVV202aydx222202()abaxdxa243abDecember2004同理，由对称性，ab1A243yVab当a=b＝R时，得球体体积，343VR球RVDecember200413yxyx课内练习求以下两曲线所围成的平面图形分别绕x轴和y轴旋转一周而成的旋转体的体积.3yx解作图12xVV312202[()]xxxd821yxwith(plots):quxian:=plot({x,x^3},x=-2..2,y=-1.5..1.5,thickness=4,color=[red,blue]):display(quxian,tickmarks=[0,0],scaling=constrained);December2004113xyxy12yVV1213202[()]dyyy81513yxyxDecember2004815yV821xVwith(plots):quxian:=plot([x,-x,x^3,-x^3],x=-1..1,y=-1.5..1.5,thickness=4,color=[red,red,blue,blue]):display(quxian,tickmarks=[0,0],scaling=constrained);December2004例圆环体的体积abTorusp.281,题16222()xyba()ab绕x轴December2004aab222ybax221ybax2221()aaVyydx224aaaaxdx222abDecember2004柱壳法(ShellMethod)题选学《高等数学学习手册》167页December2004换一种方法求体积微元x()yfxxdx22[()]Vxdxxy2[2()]xdxdxy(2)xdxy2xydxdVDecember2004ab2dVxydxbaVdV2baxydx柱壳柱壳半径柱壳的高度柱壳的厚度《高等数学学习手册》167页December2004sinyx0解例求sin(0)yxx与x轴所围成的图形分别绕x轴和y轴所得的旋转体的体积with(plots):f:=plot(sin(x),x=0..Pi,thickness=3):f1:=plot(sin(x),x=0..Pi,filled=true,color=grey):display(f,f1,tickmarks=[0,0],scaling=constrained);《高等数学学习手册》167页，例6.2.2December200420sinxVxdx22with(plots):f:=plot(sin(x),x=0..Pi,thickness=3):g:=plot(-sin(x),x=0..Pi,thickness=3):f1:=plot(sin(x),x=0..Pi,filled=true,color=grey):g1:=plot(-sin(x),x=0..Pi,filled=true,color=grey):display(f,f1,g,g1,tickmarks=[0,0],scaling=constrained);sinyx0December2004with(plots):qumian:=plot3d([x,sin(x)*cos(t),sin(x)*sin(t)],x=0..Pi,t=0..2*Pi):display(qumian,scaling=constrained);December2004with(plots):qumian:=plot3d([y*cos(t),y*sin(t),sin(y)],y=0..Pi,t=0..2*Pi):display(qumian,scaling=constrained);with(plots):f:=plot(sin(x),x=0..Pi,thickness=3):f1:=plot(sin(x),x=0..Pi,filled=true,color=grey):display(f,f1,tickmarks=[0,0],scaling=constrained);sinyx绕y轴的旋转体December2004sinyx2baVxydx02sinyVxxdx22p.28119题