Question

Use cylindrical shells to find the volume of the solid generated by rotating the curve y = 6 ln(x) about the x-axis for 1 < x < e.

Answer 1

The volume of the solid generated by rotating the curve y = 6 ln(x) about the x-axis for 1 < x < e is 2π/3.

Step-by-step explanation:

The volume of a solid generated by rotating a curve y = f(x) about the x axis between the limits x = a and x = b is given by the formula V = π∫_a^b (f(x))^2dx. In this question, f(x) = 6 ln(x). Therefore, the volume of the solid generated by rotating the curve y = 6 ln(x) about the x-axis for 1 < x < e is given by V = π∫_1^e (6 ln(x))^2dx.

Using the substitution u = lnx, we can rewrite the integral as V = π∫_0^ln(e) (6u)^2du. Solving the integral, we get V = 2π/3. Thus, the volume of the solid generated by rotating the curve y = 6 ln(x) about the x-axis for 1 < x < e is 2π/3.