Problem 24 - Thomas Calculus

Question

Write an integral for the area of the surface generated by revolving the curve \(y=\cos x,-\pi / 2 \leq x \leq \pi / 2,\) about the \(x\) -axis. In Section 8.4 we will see how to evaluate such integrals.

Step-by-Step Solution

Answer

The integral for the area is \( \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} 2\pi \cos x \sqrt{2 - \cos^2 x} \, dx \).

Step 1: Understand the Problem

We are asked to find the integral for the area of a surface generated by revolving a curve around the x-axis. The given curve is \( y = \cos x \) over the interval \( -\frac{\pi}{2} \leq x \leq \frac{\pi}{2} \). Our focus in this step is understanding that the formula for the surface area \( S \) of a curve \( y = f(x) \) being revolved around the x-axis is given by: \[ S = \int_{a}^{b} 2\pi y \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \]

Step 2: Find Derivative of y

First, find the derivative of the function \( y = \cos x \). The derivative is \( \frac{dy}{dx} = -\sin x \). This derivative helps us determine the rate of change of the function, which is crucial for setting up our surface area formula.

Step 3: Set Up the Integral Expression

Apply the surface area formula. Substitute \( y = \cos x \) and \( \frac{dy}{dx} = -\sin x \) into the formula: \[ S = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} 2\pi \cos x \sqrt{1 + (-\sin x)^2} \, dx \].

Step 4: Simplify the Expression Under the Square Root

Simplify the expression inside the square root: \( (-\sin x)^2 = \sin^2 x \), so \( 1 + \sin^2 x \).Using the Pythagorean identity \( \sin^2 x = 1 - \cos^2 x \), we get \( 1 + \sin^2 x = 2 - \cos^2 x \).Thus the expression becomes \( \sqrt{2 - \cos^2 x} \).

Step 5: Write the Final Integral

Substitute the simplified expression into the integral to get the integral for the surface area: \[ S = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} 2\pi \cos x \sqrt{2 - \cos^2 x} \, dx \].

Key Concepts

Calculus

Calculus is a branch of mathematics that studies continuous change. It is divided mainly into differential calculus and integral calculus. This area of mathematics enables us to analyze and capture the behavior of complex systems, such as calculating areas under curves, slopes of curves, and changes in functions. In particular, calculus plays a crucial role in physics, economics, engineering, and many other fields.

When exploring problems involving curves and areas, calculus allows us to employ formulas involving derivatives and integrals. These tools help in solving problems such as finding the surface area generated by revolving a curve, as illustrated with the function \(y = \cos x\). Calculus provides the framework to tackle such practical problems effectively.

Integral Calculus

Integral calculus focuses on accumulation and areas under curves. It complements differential calculus, which deals with rates of change. One common application in integral calculus is finding the surface area of a shape created by revolving a curve around a line, such as the x-axis.

To find the surface area using integrals, we apply a formula that considers the curve's length and height at each small segment. In the exercise, the formula for the surface area is:

• \[ S = \int_{a}^{b} 2\pi y \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \]

This integral expression takes into account the curve's rotation around the x-axis, ensuring accurate representation of the surface area. As we see, integral calculus is incredibly powerful for determining areas and volumes that might otherwise be difficult or impossible to find.

Pythagorean Identity

The Pythagorean identity is fundamental within trigonometry and calculus. This identity relates sine and cosine functions through the equation:

• \[ \sin^2 x + \cos^2 x = 1 \]

This relation is crucial in simplifying expressions involving trigonometric functions, especially within calculus problems.

In the given exercise, the Pythagorean identity is used to simplify the expression under the square root. By substituting \(\sin^2 x\) with \(1 - \cos^2 x\), we transform the expression \(1 + \sin^2 x\) into \(2 - \cos^2 x\). This step simplifies the integral, making it more manageable to evaluate.

Understanding and applying the Pythagorean identity helps in manipulating and simplifying integrals, making it a must-know for calculus students.