Question

The given curve is rotated about the y-axis. Find the area of the resulting surface.

y =1/4x² −1/2ln x, 3 ≤ x ≤ 5

Answer 1

To find the area of the surface resulting from rotating the curve y = 1/4x² − 1/2ln(x) about the y-axis, differentiate the equation to find dy/dx, substitute it into the surface area formula, and integrate over the given range of x values (3 to 5).

Step-by-step explanation:

To find the area of the resulting surface when the curve y = 1/4x² − 1/2ln(x) is rotated about the y-axis, we can use the formula for the surface area of a solid of revolution.

The formula is:

S = ∫2πy√(1+(dy/dx)²)dx

First, we need to find dy/dx by differentiating the equation y = 1/4x² − 1/2ln(x). After finding dy/dx, we can substitute it into the formula and integrate over the given range of x values (3 to 5). Finally, we multiply the result by 2π to get the total surface area.