Surface Area of Revolution Generated by a Curve: A Calculus Problem

In the context of advanced calculus, the surface area of a solid of revolution is a fascinating topic. Here, we will explore the calculations involved in finding the surface area of the solid generated by revolving the arc of the curve y2 2x - 1 from y 0 to y 1 around the x-axis.

Formulating the Problem

Given the equation of the curve y2 2x - 1, we can express y in terms of x as follows:

y √(2x - 1)

When this curve is revolved around the x-axis, it generates a surface of revolution. The surface area, denoted as S, can be calculated using the formula:

S 2π ∫ y ds

where ds is the differential arc length given by:

ds √(1 (dy/dx)2) dx

Calculating the Surface Area

First, let's calculate the differential arc length ds for the given curve.

(dy/dx) 1 / √(2x - 1)

ds √(1 (1 / √(2x - 1))2) dx √(1 1 / (2x - 1)) dx √(2x / (2x - 1)) dx

Substituting ds into the surface area formula:

S 2π ∫1/21 √(2x / (2x - 1)) y dx

Since y √(2x - 1), the formula becomes:

S 2π ∫1/21 √(2x / (2x - 1)) √(2x - 1) dx

Which simplifies to:

S 2π ∫1/21 √(2x) dx 2π ∫1/21 √2 x^(1/2) dx

Evaluating the integral:

2π ∫1/21 √2 x^(1/2) dx (4/3)π (x^(3/2))

Substituting the bounds of integration:

(4/3)π (x^(3/2))1/21 (4/3)π (1 - 1/(2√2)) (4/3)π (1 - 1/2√2) ≈ 1.219π

Visualizing the Solid of Revolution

The region to be revolved about the x-axis is shown in the graph below. Once completed, the revolution produces a cylinder of radius one unit, a height of one unit, and a paraboloidal concavity on its right side. The surface area of the back side of the cylinder is, of course, π square units. The curved surface of the cylinder about its circumference has an area of 2π square units. We should expect the concave surface to have an area greater than π square units, which is borne out in the calculations that follow.

For purposes of integration, we shall be concerned only with the concave surface, whose bounds of integration we take to be x 1/2 and x 1. The area of the concavity and the total surface area of the solid appear toward the end of the calculations shown in the image below. Click on that image to expand it as necessary.