The region bounded by y = e^x, y = e, and the y-axis is revolved around the y-axis. What is the volume of the resulting solid?
To find the volume of the solid formed by rotating the region bounded by the curves y = e^x, y = e, and the y-axis around the y-axis, we can use the method of cylindrical shells.
The height of each shell will be given by (e^x - e), where x is the value at which the curve y = e^x intersects y = e.
Setting e^x = e, we get x = 1.
Therefore, the height of each shell is (e^x - e) = (e - e) = 0.
The radius of each shell will be given by x, as the shells are being formed by revolving the region around the y-axis.
The volume of each shell can be given by V_shell = 2πx(e^x - e)dx.
To find the total volume, we integrate this expression from x = 0 to x = 1:
V = ∫[0,1] 2πx(e^x - e) dx
= 2π ∫[0,1] x(e^x - e) dx
= 2π [x*e^x - x*e] | [0,1]
= 2π[(1*e - 1*e) - (0*1 - 0)]
= 0
Therefore, the volume of the resulting solid is 0.