definite integration(2)

👨🏻‍🏫 學術台數學

Larry

2011-06-04 4:03 am

a)Find∫x(e^x)dxb)If theregion bounded by the curve y=e^x, the lines x=0, x=1 and the axis is revolved about x=-1, find the volume of the solid of revolution by the shell method.

更新1:

b)If the region bounded by the curve y=e^x, the lines x=0, x=1 and the axis is revolved about x=-1, find the volume of the solid of revolution by the shell method.

回答 (1)

用戶25975

2011-06-04 6:26 pm

✔ 最佳答案

a) ∫ x ex dx = ∫ x d(ex)

= xex - ∫ ex dx

= xex - ex + C

b) The vol. is:

∫ (x = 0 → 1) 2π(x + 1)ex dx

= 2π∫ (x = 0 → 1) xex dx + 2π∫ (x = 0 → 1) ex dx

= 2π [xex - ex] (x = 0 → 1) + 2π [ex] (x = 0 → 1)

= 2π [(e - e) - (0 - 1)] + 2π (e - 1)

= 2π + 2π (e - 1)

= 2πe

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