同济高等数学（第6版）习题答案9-4-1

习题9-4

1? 求球面x2?y2?z2?a2含在圆柱面x2?y2?ax内部的那部分面积?

解 位于柱面内的部分球面有两块? 其面积是相同的? 由曲面方程z?a2?x2?y2得

y?z???z??x? ? 222222?x?ya?x?ya?x?y于是 A?2 ?2x2?y2?ax??1?(?z)2?(?z)2dxdy

?x?yadxdy 222a?x?yacos?0x2?y2?ax???2 ?4a?d??01?d?22a??

2? 求锥面z?x2?y2被柱面z2?2x所割下的部分的曲面的面积?

解 由z?x2?y2和z2?2x两式消z得x2?y2?2x? 于是所求曲面在xOy面上的投影区域D为x2?y2?2x?

?4a?2(a?asin?)d??2a2(??2)?

0?y?z? 由曲面方程z?x2?y2得?z?2x2? ? 于是 22?y?xx?yx?y A?

3? 求底面半径相同的两个直交柱面x2?y2?R2及x2?z2?R2所

(x?1)2?y2?1??1?(?z)2?(?z)2dxdy?2??dxdy?2??

?x?y(x?1)2?y2?1围立体的表面积?

解 设A1为曲面z?R2?x2相应于区域D? x2?y2?R2上的面积? 则所求表面积为A?4A1? A?4??1?(?z)2?(?z)2dxdy

?x?yD ?4??1?(?Dx)2?02dxdy R2?x2 ?4??DRdxdy R2?x2?RR?x2?R12 ?4R?dx?? dy?8Rdx?16R?22?R?R?x2?RR?x 4? 设薄片所占的闭区域D如下? 求均匀薄片的质心?

(1)D由y?2px? x?x0? y?0所围成? 解 令密度为??1?

因为区域D可表示为0?x?x0, 0?y?2px? 所以 A???dxdy??dx?D0x02px0x032pxdx?22px0?

3dy??0x02pxx0111 x???xdxdy??dx?xdy??x2pxdx?3x0?

0ADA0A05x02pxx0111ydy??pxdx?3y0? y???ydxdy??dx?0ADA0A08所求质心为(3x0, 3y0) 582y2x (2)D是半椭圆形闭区域{(x,y)| 2?2?1, y?0}? ab 解 令密度为??1? 因为闭区域D对称于y轴? 所以x?0?

A???dxdy?1?ab(椭圆的面积)?

2Daa11 y???ydxdy??dx?ADA?a0ba2?x2ydy

2a1b ??2?(a2?x2)dx?4b? A2a?a3? 所求质心为(0, 4b)?

3? (3)D是介于两个圆r?acos?? r?bcos?(0?a?b)之间的闭区域?

解 令密度为??1? 由对称性可知y?0?

A???dxdy??(b)2??(a)2??(b2?a2)(两圆面积的差)?

224D22bcos?12a?b?ab? 2 x???xdxdy??d??rcos??r?dr?acos?ADA02(a?b)?22a?b?ab, 0)? 所求质心是(2(a?b) 5? 设平面薄片所占的闭区域D由抛物线y?x2及直线y?x所围成? 它在点(x? y)处的面密度?(x? y)?x2y? 求该薄片的质心? 解 M????(x,y)dxdy??dx?2xydy??1(x4?x6)dx?1?

0x0235D1x21 x?1??x?(x,y)dxdy

MD1x11113 ??dx?2xydy??(x5?x7)dx?35?

M0xM0248 y?1??y?(x,y)dxdy

MD1x111122 ??dx?2xydy??(x5?x8)dx?35?

M0xM0354