To find the volume of the solid generated by revolving the region enclosed by the curves y = √(cos(x)), y = e^x and x = pi/2 around the x-axis, we can use the method of cylindrical shells.
Consider a thin vertical strip of width dx at a distance x from the y-axis. The height of this strip is the difference between the y-coordinates of the two curves:
h = e^x - √(cos(x))
The circumference of the shell is given by 2πx since the strip is at a distance x from the y-axis. Therefore, the volume of the shell is given by:
dV = 2πx * h * dx
= 2πx * (e^x - √(cos(x))) * dx
To find the total volume of the solid, we need to integrate this expression from x=0 to x=pi/2:
V = ∫[0, pi/2] 2πx * (e^x - √(cos(x))) dx
This integral can be evaluated using integration by substitution. Let u = cos(x), then du/dx = -sin(x) and dx = du/-sin(x). Using this substitution, the integral becomes:
V = ∫[1, 0] 2π * (-ln(u)/sin(x)) * (e^x - √(u)) dx
Integrating this expression with respect to x from x=0 to x=pi/2, we get:
V = 2π * [e^(pi/2) - 1 - (4/3) * (1 - sqrt(2))]
Therefore, the volume of the solid is approximately 27.838 cubic units (rounded to three decimal places).