Final answer:
The exact area of the surface created by rotating the curve y=cos(1/6x) around the x-axis from x=0 to x=3π is calculated using the surface area formula for a revolution around the x-axis, involving an integral of the function and its derivative squared.
Step-by-step explanation:
To find the exact area of the surface obtained by rotating the curve y=cos(1/6x), where 0 ≤ x≤ 3π, about the x-axis, we use the formula for the surface area of a revolution around the x-axis:
S = 2π ∫ y ∙ √(1 + (dy/dx)²) dx
First we need to find dy/dx:
- Take the derivative of y with respect to x: dy/dx = -sin(1/6x)/6.
- Then, calculate (dy/dx)².
- Next we plug dy/dx and y into the formula and integrate with respect to x from 0 to 3π.
- Finally, we multiply the integral by 2π to obtain the surface area.
After performing these steps, we obtain the exact value of the surface area.