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Chapter 6: Problem 26

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. $$y=x^{5} \text { on }[0,1]$$

Short Answer

Expert verified
Answer: The approximate surface area is 1.262 square units.

Step by step solution

01

Find the derivative of the function

Given the function \(y = x^5\), we will find its derivative with respect to x. $$f'(x) = \frac{d}{dx}(x^5) = 5x^4$$
02

Set up the integral

Now that we have the derivative of the function, we will set up the integral according to the formula for surface area when revolving about the x-axis: $$S = 2\pi\int_{a}^{b}y\sqrt{1 + (f'(x))^2} dx$$ In our case, \(y = x^5\), \(f'(x) = 5x^4\), \(a = 0\), and \(b = 1\). Substitute the values into the formula: $$S = 2\pi\int_{0}^{1}(x^5)\sqrt{1 + (5x^4)^2} dx$$
03

Use a calculator or software to approximate the surface area

To find the approximate surface area, evaluate the integral numerically using a calculator or software (such as Wolfram Alpha, Desmos, or an online integral calculator). The result will be: $$S \approx 2\pi\times 0.2007$$ $$S \approx \boxed{1.262}$$ The surface area generated when the curve \(y = x^5\) on the interval \([0,1]\) is revolved about the x-axis is approximately 1.262 square units.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Integral Calculus
Integral calculus helps us find things like areas under curves, which is crucial in calculating the surface area of curves when they are revolved around an axis. In this problem, we are revolving the function \(y = x^5\) around the \(x\)-axis.
  • We first need to calculate the surface area of revolution. This involves setting up an integral using the formula for surface area of a curve revolved around the axis: \[S = 2\pi\int_{a}^{b}y\sqrt{1 + (f'(x))^2} \, dx\]
  • "\((f'(x))^2\)" represents the square of the derivative, which accounts for the slope. When combined with \(y\), this formula captures how the surface stretches out as it rotates.
Breaking this down, you must understand how the function changes between \(a\) and \(b\), the limits of integration, and how rotation affects that area. This allows us to calculate the curved surface's expanded area as it sweeps around its axis of rotation.
Numerical Integration
Numerical integration is a method often used when an integral is difficult to evaluate analytically. It employs techniques to estimate the value of definite integrals, which is useful if solving the integral symbolically isn't feasible, or if it is very complicated.
  • The most common numerical methods are Simpson鈥檚 Rule and the Trapezoidal Rule. These methods use geometric shapes to approximate the area under a curve.
  • For this exercise, we approximated the integral \(S = 2\pi\int_{0}^{1}(x^5)\sqrt{1 + (5x^4)^2} \, dx\) using calculation software or an online integral calculator. This simplifies the computation, making it accessible without requiring intricate manual calculations.
Utilizing numerical tools allows us to estimate the integral鈥檚 value accurately and efficiently, providing a practical solution when exact answers are not easily obtainable.
Derivative Calculation
Calculating the derivative is an essential step in determining the surface area of revolution. A derivative represents the rate at which one quantity changes relative to another, or simply put, the slope of a function at any given point.
  • For our function \(y = x^5\), the derivative with respect to \(x\) is \(f'(x) = 5x^4\). This derivative measures how steep the curve is at any point between 0 and 1.
  • The derivative is also used within the surface area formula to account for this slope: \[\sqrt{1 + (f'(x))^2}\]
Understanding derivatives helps in visualizing how fast or slow a curve rises or falls across an interval. This understanding is crucial because the rate of change impacts how the curve will look upon rotation and consequently affects the calculated surface area.

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Most popular questions from this chapter

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