Problem 48 - Calculus Early Transcendentals: Pearson New International Edition

Question

You can generate surfaces by revolving smooth curves, given parametrically, about a coordinate axis. As \(t\) increases from a to b, a smooth curve \(x=F(t)\) and \(y=G(t)\) is traced out exactly once. Revolving this curve about the \(x\)-axis for \(y \geq 0\) gives the surface of revolution with surface area $$ S=\int_{a}^{b} 2 \pi y \sqrt{\left(\frac{d x}{d t}\right)^{2}+\left(\frac{d y}{d t}\right)^{2}} d t $$ See Section 6.4. Problems 48-54 relate to such surfaces. Derive a formula for the surface area generated by the rotation of the curve \(x=F(t), y=G(t)\) for \(a \leq t \leq b\) about the \(y\)-axis for \(x \geq 0\), and show that the result is given by $$ S=\int_{a}^{b} 2 \pi x \sqrt{\left(\frac{d x}{d t}\right)^{2}+\left(\frac{d y}{d t}\right)^{2}} d t $$

Step-by-Step Solution

Answer

The surface area is given by \( S = \int_{a}^{b} 2 \pi x \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt \). This formula incorporates the required rotation around the y-axis.

Step 1: Understand the Problem Context

We are tasked with finding the surface area formula for the revolution of a parametric curve about the y-axis, given the parametric equations \(x = F(t)\) and \(y = G(t)\), with \(a \leq t \leq b\). When rotated about the y-axis, the formula should relate to the given formula for rotation about the x-axis.

Step 2: Recall the Surface Area Formula for Rotation About the y-axis

When a curve described by \(x = F(t)\) and \(y = G(t)\) is rotated about the y-axis, the differential surface area element can be expressed as \(dS = 2 \pi x \, ds\) where \(ds \) is the arc length element.

Step 3: Calculate the Arc Length Element ds

The arc length element \(ds\) for a parametric curve is given by \(ds = \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt\). This term accounts for the real-world distance along the curve as it is traced by the parameter \(t\).

Step 4: Derive Surface Area Expression

Substitute the expressions for \(x\) and \(ds\) into the surface area formula for rotation about the y-axis. This gives \[S = \int_{a}^{b} 2 \pi x \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt\]where \(x = F(t)\) and the integral sums over the interval \([a, b]\), calculating the surface area of the revolution.

Key Concepts

Parametric Curves

Parametric curves are a fascinating concept in mathematics that allow us to describe curves through a set of equations. Instead of using the traditional form of expressing a curve as a function of y in terms of x or vice versa, parametric equations use a third variable, usually denoted as \(t\), called the parameter. This variable progresses through a specific interval and traces the path of the curve in a coordinate plane. For example, for a curve described by:

• \(x = F(t)\)
• \(y = G(t)\),

as \(t\) increases from a to b, the ordered pairs (x,y) create the desired curve.Using parametric equations can make it easier to work with curves that might be difficult to represent with a single equation in standard Cartesian form, such as curves that loop back on themselves or have vertical tangents.

Arc Length

The arc length is an essential concept when dealing with curves, as it provides us with an understanding of the true length of a curved path. For parametric curves, the arc length element \(ds\) is a critical component, especially when calculating things like surface areas or distances.To find the arc length of a parametric curve, use the formula:\[ds = \sqrt{\left(\frac{dx}{dt}\right)^{2} + \left(\frac{dy}{dt}\right)^{2}} \ dt\]This formula evaluates the small distance (arc length) changes along the curve as the parameter \(t\) varies. By integrating \(ds\) over the interval \([a, b]\), you obtain the total arc length of the curve over that parameter range.

Integral Calculus

Integral calculus is the branch of mathematics focused on accumulation, which includes finding areas under curves, volumes, and in this case, surface areas when dealing with surfaces of revolution. In our context, the integration process is essential for summing up infinitesimally small elements (such as arc lengths or areas).The integration is often expressed in the form of an integral:\[S = \int_{a}^{b} f(x) dx\]For the surface of revolution generated by a parametric curve, this integral sums up all of the infinitesimal surface area elements around the axis of revolution. Here, integral calculus allows us to compute the total surface area accurately and systematically.

Surface Area Formula

The surface area formula for a surface of revolution is an application of integral calculus. This formula helps calculate the surface area generated when a curve is rotated around a coordinate axis.For a curve defined parametrically by \(x=F(t)\) and \(y=G(t)\), revolving around the y-axis translates to using the formula:\[S = \int_{a}^{b} 2 \pi x \sqrt{\left(\frac{dx}{dt}\right)^{2} + \left(\frac{dy}{dt}\right)^{2}} \, dt\]This expression combines several elements:

• The \(2 \pi x\) factor accounts for the circumference of the circle made by rotating the curve piece at x around the y-axis.
• The arc length element \(ds\) ensures that the distance along the curve is captured accurately.

By integrating this expression over the parameter interval \([a, b]\), we obtain the total surface area of the generated solid.