求曲線y=e^(-x)與直線x=0,x=1,y=0所圍成的平面圖形繞Y軸旋轉(zhuǎn)一周而成的旋轉(zhuǎn)體的體積

數(shù)學(xué)人氣：798 ℃時(shí)間：2019-11-06 15:37:59

優(yōu)質(zhì)解答

x = 0,y = e^0 = 1
x = 1,y = 1/e
繞y軸旋轉(zhuǎn),用y做自變量較方便:y = e^(-x),x = -lny
0 < y < 1/e時(shí),旋轉(zhuǎn)體為:截面為半徑=1,高為1/e的圓柱,體積V1 = π*1²*1/e = π/e
1/e < y < 1處,旋轉(zhuǎn)體截面為以|-lny|為半徑的圓,V2 = ∫πl(wèi)n²ydy
= πy(ln²y - 2lny + 2) (1/e ->1)
= π(0 - 0 +2) - π(1 + 2 + 2)/e
= 2π - 5π/e
V = V1 +V2 = π/e + 2π - 5π/e
= 2π - 4π/e查表，哈哈。