To set up the integral using the method of shells, we need to consider small cylindrical shells that are formed by rotating a vertical line segment around the specified axis.
Given curves:
y = e^x,
x = 0,
y = 2
Axis of rotation:
y = 1
First, we need to find the limits of integration for the variable used in the integral. In this case, we will integrate with respect to y.
To find the limits of integration for y, we need to determine the y-values at which the curves intersect the axis of rotation (y = 1).
For the curve y = e^x:
e^x = 1
x = ln(1) = 0
For the curve x = 0:
y = e^0 = 1
Therefore, the limits of integration for y are from y = 1 to y = 2.
Next, we consider a small vertical strip with width Δy (delta y) and height h. The height h can be calculated as the difference between the y-values of the curves, i.e., h = (2 - e^x).
The radius of the shell is the distance from the axis of rotation (y = 1) to the curve y = e^x, which is given by r = (e^x - 1).
The volume of the small cylindrical shell is approximately given by:
dV = 2πrh Δy,
where r represents the radius, h represents the height, and Δy represents the width of the shell.
Therefore, the integral for the volume of the solid obtained by rotating the region bounded by the curves y = e^x, x = 0, y = 2, about the axis y = 1 is:
V = ∫(1 to 2) 2π(e^x - 1) * (2 - e^x) dy