Problem 45 - Essential Calculus

Question

In Exercises 43-46, find the area of the surface formed by revolving the curve about the given line. $$ \begin{array}{lll} \underline{\text { Polar Equation }} & \underline{\text { Interval }} & \underline{\text { Axis of Revolution }} \\ r=e^{a \theta} & 0 \leq \theta \leq \frac{\pi}{2} & \theta=\frac{\pi}{2} \end{array} $$

Step-by-Step Solution

Answer

The surface area of the surface formed by revolving the curve \(r = e^{a\theta}\) about the line \(\theta = \(\frac{\pi}{2}\) is given by \( A = 2\pi\int_{0}^{\frac{\pi}{2}} e^{a\theta}\sqrt{1+(ae^{a\theta})^2}d\theta\). This integral cannot be solved exactly using elementary functions.

Step 1: Compute the Derivative

Differentiate the function \(r(\theta) = e^{a\theta}\) to find \(r'(\theta)\). Using the chain rule, \(r'(\theta) = ae^{a\theta}\).

Step 2: Substitute \(r(\theta)\) and \(r'(\theta)\) into the Formula

Substitute the original function and its derivative into the formula for the surface area of a revolution in polar coordinates. This gives \( A = 2\pi\int_{0}^{\frac{\pi}{2}} e^{a\theta}\sqrt{1+(ae^{a\theta})^2}d\theta\). This integral represents the surface area of the revolution.

Step 3: Simplify the Integral

This integral cannot be solved using elementary functions. However, it can be left as is, or numerically approximated if the value of a is known.

Key Concepts

Polar Equations

Polar equations are mathematical expressions that describe curves on a plane using polar coordinates instead of the traditional Cartesian coordinates. In polar coordinates, each point on the plane is determined by a distance from the origin, denoted as the radius r, and an angle θ from the positive x-axis, also known as the polar axis.

Polar equations take the form r(θ) = f(θ), where the function f provides a rule for computing the radius for any given angle. Common shapes like circles, spirals, and roses have simple representations in polar form. For instance, a spiral can be represented by the exponential function r(θ) = e^aθ, where a is a constant that affects the tightness and direction of the spiral.

When dealing with surface areas of revolution in polar coordinates, these equations allow us to describe the curve that is being rotated about an axis. The key to working with polar equations is understanding the relationship between r and θ and how they define the position of a point in the plane.

Calculus Integration

Calculus integration is a fundamental concept used to calculate areas, volumes, and other quantities that accumulate over a continuous range. It works by summing an infinite number of infinitesimally small quantities. When the surface area of a figure of revolution is sought, calculus integration becomes particularly handy.

In the context of polar equations, integration is used to find the total area or surface area generated by a curve as it revolves around an axis. The integral takes the form of an area under the curve in a polar coordinate system. For the surface area of a revolution, the integral will typically involve r(θ) and its derivative r'(θ).

The general formula for the surface area of a revolution in polar coordinates is A = 2π∫r(θ)&sqrt;{1+[r'(θ)]²}dθ, which accounts for infinitesimal bands of surface area around the axis as the curve rotates. This compels the student to set up and evaluate these integrals, often leading to complex and interesting calculus problems.

Chain Rule Differentiation

The chain rule is a powerful differentiation tool in calculus that allows us to find the derivative of composite functions. When a function u is composed with another function v, u(v(θ)), the derivative of this composite function with respect to θ is given by u'(v(θ))×v'(θ).

This rule is integral when finding the derivative of polar equations like r(θ) = e^aθ with respect to θ. Applying the chain rule, the derivative r'(θ) is computed as ae^aθ, where a is treated as a constant multiplier. The derivative r'(θ) is then used in the surface area formula, highlighting the importance of understanding and applying the chain rule.

Chain rule differentiation facilitates the step-by-step process in solving complex calculus problems, such as finding the surface area of shapes created by revolving curves defined in polar coordinates. It is essential for students to grasp this concept to successfully maneuver through calculus-related exercises and tasks.