The given curve is rotated about the y-axis. Set up, but do not evaluate, an integral for the area of the resulting surface by integrating (a) with respect to x and (b) with respect to y.x = e3y,    0 ≤ y ≤ 2(a)Integrate with respect to x.

The surface area of a solid of revolution can be found using the formula:

A = ∫2πf(x)√[1 + (f'(x))^2] dx

where f(x) is the function being revolved, and f'(x) is its derivative.

(a) Integrate with respect to x:

First, we need to express the function in terms of x. Given x = e^(3y), we can take the natural logarithm of both sides to get y = ln(x)/3.

The derivative of y with respect to x is then 1/(3x).

Substituting these into the formula gives:

A = ∫2π(e^(3y))√[1 + (1/(3x))^2] dx

This is the integral you would evaluate to find the surface area when integrating with respect to x.

(b) Integrate with respect to y:

When integrating with respect to y, we can use the function as given, x = e^(3y).

The derivative of x with respect to y is then 3e^(3y).

Substituting these into the formula gives:

A = ∫2πy√[1 + (3e^(3y))^2] dy

This is the integral you would evaluate to find the surface area when integrating with respect to y.