19.求心脏线r?a(1?cos?)的全长. 解r??a(?sin?) ?22 s?2a?0(1?cos?)?sin?d?? ????8a. ?4a?0cosdx?8asin220
20.试证双纽线r2?2acos2?(a?0)的全长L可表为L?422a?10dx1?x4.20
22证2rr???4asin2?,r???2asin2?/r,
42?/44asin2? 2s?4?2acos2??d?20 2acos2? ?/41?42a?d? 0cos2? ?/4d??42a ?022cos??sin?
?/4d? ?42a?02222(cos??sin?)(cos??sin?)
?/4 d??42a?440 cos??sin? ?/4dtan??42a(tan??x) ?041?tan?
1dx ?42a.?041?x
21.求抛物线y?1?解 y??x220x24(0?x?2)绕x旋转所得的旋转体的侧面积. .?x??x?1?1?????dx4??2??322S?2?14?20???4?xn2dx?4???/40dxcosx5
In??secxdx?n?2?secn?2xdtanxn?2?sec?secIn?xtanx??(n?2)secxtanxdx2n?2xtanx?(n?2)In?(n?2)In?2,secn?21n?1xtanx?n?2n?1In?2. 6
I15?4sec3xtanx?34I33?14secxtanx?3?11?4?secxtanx?I?221??
?1sec3xtanx?3348secxtanx?8ln(tanx?secx)?C.S?4?(1sec3xtanx?348secxtanx?38ln(tanx?secx))|?/40
??2[72?3ln(1?2)].2222.求x旋转而成的椭球面的面积a2?y21b?(0?b?a)分别绕长,短轴.解?x?acost?y?bsint,0?t?2?,x???asint,y??bcost.?S?/2222a?2?2?b?0asint?bcos2tsintdt???4?b?/2?a2?(a2?b2)cos20tdcost?1?4?b?a20?(a2?b2)u2du1?4?ba2?b2?0??2?u2du2?4?aba?b2?u??21a???2?u2?2arcsin?u??2??0?2?ab??1??2?arcsin?????.?S?/2222b?2?2?a?s20asint?bcotcostdt?4?a?/2?b20?(a2?b2)sin2tdsint1?4?a?b20?(a2?b2)u2du2?4?aa2?b21?b0a2?b2?u2du1?2b?b22?4?aa2?2ub?22?u2?22?2a?b2(a?b)ln(u?b2?a2?b2?u)???02?2?a2?2?ba?ln???b(1??)??.?23.计算圆弧x2?y2?a2(a?h?y?a,0?h?a)绕y轴
旋转所得球冠的面积.7
?x?acosta?h? ?解arcsin?t?.a2 ?y?asint?a 222S?2??xx??y?dta?harcsin
aa?h ?2? 2?a?2a?hcostdtarcsina
?2 ??a?sint?2a?harcsin a a?h?2??2?a1??2?ah.?? a??
24.求心脏线r?a(1?cos?)绕极轴旋转所成的旋转体的侧面积. 解r?=-asin?. ?2222 S?2??0a(1?cos?)sin?a(1?cos?)?asin?d?? 23/2?2?a2?(1?cos?)sin?d?0
? ??2?a22?1?cos??3/2dcos??0 123/2?2?a2(1?x)dx ??1 1225/2?2?a2(1?x) ?15 322??a.
525.有一细棒长10m已知距左端点x处的线密度是?(x)?(7?0.2x)kg/m求这细棒的质量.解m??100?(7?0.2x)dx???7x?0.1x?210?80(kg).026.求半径为a的均匀半圆周的重心坐标.?x?acost解由对称性,x0?0.?,0?t??y?asint?y0???0asintadt?a2a?).a?[?cost]|0??2a?.重心坐标(0,?27.有一均匀细杆,长为l.质量为M.计算细杆绕距离一端l/5处的转动惯量.解??M/l.J?3l/5?l/50Mlxdx?2?4l/50Mlxdx2?Mxl3?0Mxl34l/53?01375Ml.
2 8
28.设有一均匀圆盘,半径为a,质量为M,求它对于通过其圆心且与盘垂直的轴之转动惯量.解??a0
.dm=MM?ax22?a?22?xdx?4a2Mxdxa122.
J?
?2Mxdxa22Mxa2
Ma.
24?029.有一均匀的圆锥形陀螺,质量为M,底半径为a,高为h,试求此陀螺关于其对称轴的转动惯量.
解y?ahx,??M1?3M?a2h,dm????a2?x?3M?dx?2h3xdx
3?a2h?h? dJ?1?a?213a2dmM?x4?h??xdx 2?2h5 h225h
J??13aM413aMx02h5xdx?2h55?3Ma2.
01030.楼顶上有一绳索沿墙壁下垂,该绳索的密度为2kg/m.若绳索下垂部分长为5m,求将下垂部分全部拉到楼顶所需做的功.解dW?2?9.8xdx.
5W???25 029.8xdx?9.8x0?25?9.8(J).
31.设y?f(x)在[a,b]上连续,非负,将由y?f(x)x?a,x?b及x轴围成的曲边梯形 垂直放置于水中,使y轴与水平面相齐,求水对此曲边梯形的压力. 解dS?f(x)dx,dF??pdS?g?xf(x)dx,
?bF?g?axf(x)dx. 32.一 水闸门的边界线为一抛物线,沿水平面的宽度为48m,
最低处在水面下64m,求水对闸门的的压力.24 解y?64?ax2,0?64?a?242,a?1,x??3(64?y). 9 ?64F?6g?0y64?ydy.64?y?u,y?64?u2, y?0时u?8,y?64时,u?0.8 F?6g??0(64?u2)u(2u)dy 38 ?12g??uu5??64??5??52428.8g?. ?3?640 9