Final answer:
To find the volume of the solid generated by rotating the region bounded by the curves y=eˣ, y=e⁻ˣ, and x=1 about the y-axis using the method of cylindrical shells, we need to integrate the volume of each cylindrical shell.
Step-by-step explanation:
To find the volume of the solid generated by rotating the region bounded by the curves y=eˣ, y=e⁻ˣ, and x=1 about the y-axis using the method of cylindrical shells, we need to integrate the volume of each cylindrical shell. The radius of each shell is the distance from the y-axis to the curve, which can be expressed as eˣ - e⁻ˣ. The height of each shell is the difference in y-values between the curves, which is eˣ - e⁻ˣ.
So, the volume of each shell is given by V = 2π(eˣ - e⁻ˣ)(eˣ - e⁻ˣ)dx. To find the total volume, we integrate this expression from x=0 to x=1: ∫012π(eˣ - e⁻ˣ)(eˣ - e⁻ˣ)dx. Evaluating this integral will give us the volume of the solid generated by the rotation.