习题3.
求下列曲线所围成的的图形的面积: 1.y?x2与x?y2.2
解求交点:??y?x?,x?x4,??x?y2 x(1?x)(1?x?x2)?0,x?0,x?1. 1S??213/20(x?x2)dx????1x2??1.?3x3??0322.y?x,y?1与y?x4.131解S??0(y?2y)dy??y2???y2??3?2?2.03.y2?2x?1与x?y?1.2解?y?2x?1?x?y?1(x?1)2?2x?1,?x2?4x?0,x?0,y??1;x?4,y?3. 3S????1?y?1?1(y2?1)??2?dx?23??3y?1?y?y3??16.?226???13
4.y?0与?x?a(t?sint)?y?a(1?cost) 0?t?2?(a>0)? 2S???0a(1?cost)da(t?sint) ?a22??20(1?cost)dt22 ?4a??sin4t402dt?8a2??0sinudu ?16a2?/2?420sinudu?16a34?2??2
?3?a2.5.y?x2?4与y??x2?2x.?y?x2解??4 ?x2?4??x2?2??y??x2x,?2x
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2x2?2x?4?0,(2x?2)(x?2)?0,
x1??2,x2?1.1S???2(?x2?2x?x2?4)dx 1??2?2(?2x?2x?4)dx ?21???x3?x2?4x??9. ?3???2
6.x2?y2?8与y?12x2(分上下两部分).22 ?x?y?8解??1x2?1x4?8
?24?y?2xx4?4x2?32?0,x2?u u2?4u?32?0,(u?8)(u?4)?0u2 2??8(舍)u2?4,x?4,x1??2,x2?22S1?
???2?8?x2?12?2x??dx?221 ?2??0?8?x?2x2???dx?2 ?2?x2?8?x?4arcsinx??2??4?222??03 S2?8???4?2???4?3??6??.?3 7.y?4?x2与y?x?2. 解??y?4?x2y?x?24?x2?x?2
?x2?x?2?0,(x?2)(x?1)?0, x1??2,x2?1.
S?1?22?2(4?x?x?2)dx?6?1??2(x?x)dx 321?6??x??x?2??9. ?3??22
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8.其求双纽线r?acos2?(a?0)所围图形的面积.1解S=4?22??/41?/42acos2?d??2a?sin2?|0?a.22
202
求下列曲线围成的平面图形绕轴旋转 所成旋转体的体积:
9.x2/3?y2/3?a2/3(a?0). 解??x?acos3t ?,0?t??y?asin3?2?.t ?aV?2?y2dx?2??/2?2620asinta?03costsintdt ?6?a3?/2?720sintcostdt ?6?a3?/2?sin7t(1?sin20t)dt ?6?a3?6?4?2?8??7?5?3?1???32
?9??105?a3.???1
0.y?ex?1,x?ln3,y?e.ln3(exln3????1)2x
Vdx??2ex?1)dx0??(e20ln3 ???1?e2x?2ex?x????ln3.?2?0
求下列平面曲线围成的平面图形绕轴 旋转所成旋转体的体积:23
11.ay?x,x?0及y?b(a?0,b?0).解x?a1/3y2/3,
bV???1/3y2/30(a)2dy ??a2/3b?3y7/3?3/3707?a2b7/3.
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12.x?8lnyy,x?0,y?e.e8lny
解V???1y2dye ?8????lnydy?1???1??e
?8?????y?1lny|e?11??1ydlny???e
?8???2??1?dy??e?1y?? ?8??1?1e????e?y1??8????1?2?.?e??13.设y?f(x)在区间[a,b](a?0)上连续且不取负值，试用微元法推导：由曲线y?f(x)，直线x?a，x?b及轴围成的平面图形绕y轴旋转所成立体的b体积为V?2??xf(x)dx.
ab解厚度dx的圆筒的体积dV?2?xf(x)dx,V?2??xf(x)dx.a x
14.求曲线y?e,x?1,x?2及x轴所围成的平面图形绕y轴旋转所成的立体的体积. 2解V?2??1xexdx?2??2x???1xde??? 2?2??2??xexex2?1??1dx????2??2x??2e?e?e1?? ?2??2?2e?e?(e2?e)?2??2?e.
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15.证明:半径为a高为h的球缺的体积为
V??h2??h?a??3?.? 证y?f(x)?a2?x2,a?h?x?a.a
V???2a?h(a?x2)dx???213a??ah?x|?3a?h?? ???2133?2?h??ah?(a?(a?h))??h
?3???a??3?? 3 16.求曲线y?x6?1在2xx?1到x?3之间的弧长. x21 解y???. 22x2 322s??x1? ?11????22x2?dx? 3?x4?1?x331
?12x2dx??14???62x??.?13
17.求曲线r?asin3?的全长.
3 解r??asin2?cos?
33, 3s?a?
?sin6?03?sin4?3cos2?3d? 3
?a??0sin2?3d? ?6a?/2?21?30sin?d??6a?2?2?2?a.
18.求向星形线x?acos3t,y?asin3t的弧长.
解x??3acos2t(?sint),y??3asin2tcost 4?/2s??3a?sintcostdx?12a?1sin2t|?/2?6a. 020
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