第二十五次课 20土木工程1、2班 75（101）人 ② 定积分的几何应用计算体积 ③ 定积分的几何应用计算弧长 -------------------------+-
?dx
1、平行截面面积可知的立体的体积 2、计算旋转体的体积
V??A?x?dx A?x?是截面面积函数.
ab
A(x)??1R2?x2tan?R2?x2212R?x2?tan??2R1V?2??R2?x2?tan?dx02?tan??R0
?R2?x2?dx?R?2x3?2?tan??Rx???R3tan?3?03?
A?x??hR2?x2V?2?hR2?x2dx0R
?2h?R0R?xdx?22?R2h2函数曲线y?f?x?,x??a,b? 绕X轴旋转的旋转体的体积：
V???fab2?x?dx???aby2dx
函数曲线y?f?x?,x??a,b?绕Y轴旋转的旋转体的体积：
V?2??xf?x?dx?2??xydx
aabb函数曲线x??(y)(c?y?d)绕Y轴旋转的旋转体的体积：
V????2(y)dy
cd函数曲线x??(y)(c?y?d)绕X轴旋转的旋转体的体积：
V?2??yxdy?2??y??y?dy
ccdd-------------------+-
x2y2??1 a2b2绕x轴
x2y2b2222??1?y?a?x??a2b2a2ab2a22V?2??ydx?2?2??a?x2?dx
0a0b2?2?2a?2x3?4ax???ab2??3?03?a4?ba2绕Y轴V?3
x2y2z2计算椭球体2?2?2?1的体积？
abc
x2y2z2y2z2x2???1?2?2?1?2?a2b2c2bcay2x2?2?b?1?2?a???z2x2?2?c?1?2?a???1??x2?x2?x2?2?A?x???b?1?2??c?1?2???bc?1?2?a?a?a????2
V=2?a0?x2?A?x?dx?2?bc??1?2?dx0a??aa?x3?4?2?bc?x?2???abc3a?03?
如果函数曲线用参数习惯表示
??x???t?????t???,?? ???y???t?绕X轴旋转的立体的体积
??x???t????y???t?V???ydx???2abab??1?b??1??a??2?t????t?dt???绕X轴???t???t????t?dt???绕Y轴??
V?2??xydx?2????1?b???1?a???x?a?t?sint?(a?0) 的一拱与X轴所围成的图形绕X、Y轴旋转而计算摆线???y?a?1?cost?成的立体的体积.
??x?a?t?sint?(a?0)???y?a?1?cost?V?V1?V2???2a03x1dy???22a02x22dysintdt??a3
??a???x?a?t?sint?(a?0)???y?a?1?cost?V?????a3?2??t?sint???0?t?sint??2sintd2?a02?ydx???a?1?cost?dt232?30?2t?2sin?dt?0?2??2?3tt?8?a?sin6dt???????u,dt?2du022?16?a?32?a33
??0sinudu?32?a63??20sin6udu35??5?2a332??x?a?t?sint?(a?0)?y?a1?cost????V?2???2?a?2?a?2?a32?a02?xydx23?0?t?sint??1?cost?dt2?2t?sint1?2cost?cost?dt????0??t?2tcost?tcos02?2
3t?sint?sin2t?sintcos2t?dt
绕Y轴
?x?a(t?sint) ?y?a(1?cost)?V?2??2?02?a0xf?x?dx?2??2?a0xydx?2??a(t?sint)a2(1?cost)2dt ?2?a3?2?0t(t?sint)(2sin2)2dt2?8?a3??16?a32?0(t?sint)sin4t?t?dt??????u,dt?2du?2?2???0???(2u?sin2u)sin4udu????v?u??2??????16?a3?2?(2v???sin?2v???)sin4?v??du?2? ?2?16?a3?2?(2v???sin?2v?)cos4vdu?2?????16?a?6?3a3
23??2?cosvdu?32?a4232?20cos4vdu
2024年高等数学课件-第二十五次课 20土木工程1、2班 75(101)人
