積分的計算及應用第三章習題

求下列曲線所圍成的的圖形的面積1.yx2與xy2

解求交點x2xxxy x(1x)(1xx2),2x0,x11. S x2)dx

x3/2

x21 yxy1與y41 解S

(y y)dy

y2 y22x1與xyy22x解xy

(x1)22xS y (y21) y y

y3

16 ya(1cosy0與ya(1cos

0tS0a(1cost)da(tsina22(1cos0 4a2 4tdt8a2 4 16a2/2sin4udu16a20

42yx24與yx222yx

x24x22x22x40,(2x2)(x2)x12,x2S1(x22xx21(2x22x2 x3x24x13 3 x2y28與y1x2分上下兩部分2 y

x24x4x44x2320,x2u24u320,(u8)(u4)u8(舍)u4,x24,x2,x 2 12 S 8 28222 12808 x8x8

4 x22 2 S

2464 3 3 3 7.y4x2與yxy4解yx

4x2xx2x20,(x2)(x1)x12,x2 S1(4x2x2)dx61(x2 x2 其求1/

r2a2cos2(a0)所圍圖形的面積解S= 2 所成旋轉體的體積:x2/3y2/3a2/3(axacos3解 yasin

,0t V2ay2dx2/2a2sin6ta3cos2tsin 06a3/2sin7t(1sin20 10.yex1,xln3,y Vln3(ex1)2dxln3(e2x2ex 22

e2x2

ln

ln旋轉所成旋轉體的體積:ay2x3x0及yb(a0,b解xa13y23Vb(a1/3y2/30a2/33y7/3b3a2/3b7/3 8lnx ,x0,y8lnyln 8elnydy 81e y1e 2e

1

曲線y

f(x)在區(qū)間[ab](a0)b體積為V2axfb 求曲線yexx1,x2及x軸所圍成的平 解V2xexdx2xdex 2xex22exdx22e2eex2 122e2e(e2e)證明:半徑為a高為hV2

h3h 3 證yf(x) a2x2,ahxaV (a2x2 a2h1x3|aa3 ah3ha2h1(a3(ah)3)h2ah 求曲線yx3

21解y 11x2x2s 3x x11412x2dx 2x 3 求曲線rasin3 3解rasin2 sa

sin6

2 da3sin2d 6a/2sin2d6a1 2 求向星形線xacos3t,yasin3t的弧長解x3acos2t(sint),y3a1sin2tcoss /2sintcostdx12asin2t|/2 求心臟線ra(1cos)的全長 解r s2a0

(1cos)sin4acosx8asind 2試證雙紐線r22a2cos2(a0)的全長L可表為L證2rr4a2sin2r2a2sin2

.112acos4s2acos4142a/ 142a/ 42a/ 042a/ cos4sin442a/ dtan(tan142a 102求拋物線y2解yx2x2x

x(0x2)繞x旋轉所得的旋轉體的側面積.S2014 1

23

4xdx cos5In xdx 2xdtan 2xtanx(n 2xtan2 2xtanx(n2)In(n2)In2I

2xtanxn2 n

n1 1I1sec3xtanx3I1sec3xtanx31secxtanx 1 44 1sec3xtanx3secxtanx3ln(tanxsecx) S4 3secxtanx3ln(tanxsecx))|/(4

xtan 2

求

1(0ba)分別繞長,短軸旋轉而成的橢球面的面積.ybsin解xacost0t2xasintybsina2a2sin2tb2cos2Sa

0

011

a2(a2b2)cos2tdcosta2(a2b2)u2a2a2

a2 a2 2u2

cos011

b2(a2b2)sin2tdsinb2(a2b2)u2a2a211aa2a2

b2(a2b2

ln(u

lnb計算圓弧x2y2a2ahya0ha)繞y軸解xacostarcsinahtyasin 2S2

xxxx2y22a2

ahcosaa2sint

2a2ah

a 求心臟線ra(1cos)繞極軸旋轉所成的旋轉體的側面積解rS

a(1cos)sina(1cos)a0

2(10

5

21cos3/2d021(1x)3/0 有一細棒長10m已知距左端點x處的線密度是x)(70.2x)kg/m解m10(70.2x)dx7x0.1x210 xacos解由對稱性,x00.yasint0tasiny

2a 重心坐標(0,2a有一均勻細桿,長為l.質量為M.計算細桿繞距離一端l/5處的轉動慣量解Ml.Jl/5Mx2dx4l5M M M 0

4l/MM

Ml設有一均勻圓盤,半徑為a質量為M求它對于通過其圓心且與盤垂直的軸之解M.dmM2xdx2Mxdx

2Ma2MaJ

a2x2xa

1Ma2 有一均勻的陀螺,質量為M,底半徑為a,高為h,試求此陀螺關于其對稱軸的2解yax, 3M,dm xdx3Mx2

a3

a h dJ1dmax21

h 5 55J

h13aMx4dx13aM

3Ma20

解dW29.8xdx.W529.8xdx9.8x25259.8(J 設yf(x)在[ab]上連續(xù),非負將由yf(x)xaxb及x軸圍成的曲邊梯形求水對此曲