Problem 26 - Calculus Early Transcendentals

Question

Surface area using technology Consider the following curves on the given intervals. a. Write the integral that gives the area of the surface generated when the curve is revolved about the \(x\) -axis. b. Use a calculator or software to approximate the surface area. $$y=x^{5} \text { on }[0,1]$$

Step-by-Step Solution

Answer

Answer: The approximate surface area is \(0.669\) square units.

Step 1: Finding the Derivative of y with respect to x

To find the derivative of \(y=x^5\), we will use the power rule which states that if \(y=x^n\), then \(\frac{dy}{dx} = nx^{n-1}\). So, the derivative of \(y\) with respect to \(x\) is: $$ \frac{dy}{dx} = 5x^{4} $$

Step 2: Setting up the Integral for Surface Area

We now have our function \(y=x^5\) and its derivative \(\frac{dy}{dx}=5x^4\). We can now set up the integral for the surface area: $$A = 2\pi\int_{0}^{1} x^5 \cdot \sqrt{ 1 + (5x^4)^2 } dx$$

Step 3: Approximating the Surface Area

Now, we will use a calculator or software to approximate the value of the integral: $$A \approx 2\pi\int_{0}^{1} x^5 \cdot \sqrt{ 1 + (5x^4)^2 } dx \approx 0.669$$ So, the surface area of the curve \(y=x^5\) on the interval \([0,1]\) when it is revolved about the \(x\)-axis is approximately \(0.669\) square units.

Key Concepts

Integral Calculus

Integral Calculus is a branch of mathematics focused on the accumulation of quantities and the areas under and between curves. When you're working with integrals, you're essentially summing infinitely small pieces of data to find whole quantities. This is crucial in calculating areas, volumes, and even the surface areas of three-dimensional shapes formed through functions. In this particular exercise, we are using integral calculus to determine the surface area created when a function is revolved around the x-axis.
Integrals come in different forms: definite and indefinite. A definite integral provides a numerical value representing the area under a curve between two points, while an indefinite integral represents a family of functions. In our case, we are dealing with a definite integral to find a specific value for the surface area on the interval \([0,1]\). Utilizing integration technique gives us the exact measurement needed to solve such real-world problems in mathematics.
Understanding this concept allows you to tackle similar surface area problems efficiently, providing you with a solid foundation for more advanced calculus applications.

Revolution about x-axis

Revolution about the x-axis is a technique used to generate three-dimensional shapes from two-dimensional functions. By revolving a curve around the horizontal x-axis, we can form unique and complex shapes, and the task is to calculate properties such as volume or surface area of these shapes.
In our exercise, we take the function \(y = x^5\) and consider its revolution about the x-axis over the interval \([0,1]\). This generates a surface, and we want to find the total area of this surface. The calculus technique to find the surface area of revolution involves integrating a specific formula.
The formula for the surface area \(A\) is:

• \(A = 2\pi\int_{a}^{b} y \cdot \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx\),
• where \(y\) is the function being revolved and \(\frac{dy}{dx}\) is its derivative.

This integration considers the curve's length and wraps it around the circular path created by the revolution.

Power Rule

The Power Rule is a fundamental tool in calculus used to find derivatives of functions in the form \(y = x^n\). It simplifies the differentiation process considerably and is one of the first rules one learns in calculus.
According to the Power Rule, the derivative of \(y = x^n\) is given by \(n \cdot x^{n-1}\). This means you multiply the power \(n\) by the term itself minus one from the exponent. In the case of our function \(y = x^5\), applying the Power Rule, the derivative is:

• \(\frac{dy}{dx} = 5x^4\).

This derivative is essential for setting up integrals related to surface area calculations, as it reflects the curve's slope or rate of change needed for later integration.
Mastering the Power Rule makes solving more complex calculus problems much easier as it forms the basis of further derivative rules and integration processes.

Approximating Integrals

Approximating integrals is a crucial skill, especially when dealing with complicated functions that are difficult or impossible to integrate analytically. Tools like calculators and computer software render numerical methods for approximating the solution to integrals.
In our exercise, to find the exact surface area, we use a numerical method to approximate the integral:

• \(2\pi\int_{0}^{1} x^5 \cdot \sqrt{1 + (5x^4)^2} dx\).

Numerical integration can be achieved through various methods including Simpson’s Rule, Trapezoidal Rule, and others, depending on the function's complexity.
For the given integral, the approximation yields a surface area of approximately \(0.669\) square units. Approximations allow us to gain insight into values we could not otherwise solve by standard algebraic means, thus bridging the gap between theoretical calculus and practical application.