Question

Find the exact area of the surface obtained by rotating the curve about the x-axis. y² = x, 1 ≤ x ≤ 3?

Answer 1

The exact area of the surface obtained by rotating the curve y² = x about the x-axis from x = 1 to x = 3 is calculated using the integral for the surface area of revolution, incorporating the differential arc length and its derivative.

Step-by-step explanation:

To find the exact area of the surface obtained by rotating the curve y² = x about the x-axis between x = 1 and x = 3, we can use the formula for the surface area of a revolution around the x-axis, which is provided by the integral:

S = 2π∫ y ∙ ds

where ds = ∙ dx. Here ds is the differential arc length of the curve defined as ds = √(1 + (dy/dx)²)dx.

First, we compute dy/dx:

1. Since y² = x, differentiating both sides with respect to x gives us 2y ∙ dy/dx = 1.
2. So, dy/dx = 1/(2y).

Next, we substitute dy/dx into the ds expression:

1. ds = √(1 + (1/(2y))²)dx.
2. ds = √(1 + x/(4x²))dx because y² = x.
3. ds = √(1 + 1/(4x))dx.

Now, we can set up our integral:

1. ds = √(1 + 1/(4x))dx.
2. S = 2π∫² ∙ √(1 + 1/(4x))dx from x = 1 to 3.
3. The computation of this integral will give us the exact surface area.