Question

Use a computer algebra system to solve the following problems. Find the approximate area of the surface generated when the curve \(y=1+\sin x+\cos x,\) for \(0 \leq x \leq \pi,\) is revolved about the \(x\) -axis.

Short answer

Answer: To find the approximate area, follow these steps: 1. Calculate the first derivative of the curve with respect to \(x\), which is given by \(\frac{dy}{dx} = \cos x - \sin x\). 2. Square the derivative and add 1: \((\cos x - \sin x)^2 + 1 = \cos^2 x - 2\sin x\cos x + \sin^2 x + 1\). 3. Find the square root of the result in step 2: \(\sqrt{\cos^2 x - 2\sin x\cos x + \sin^2 x + 1}\). 4. Multiply the curve by the result of step 3: \((1+\sin x+\cos x)\sqrt{\cos^2 x - 2\sin x\cos x + \sin^2 x + 1}\). 5. Integrate the expression from step 4 over the interval [0, \(\pi\)]: \(\int_{0}^{\pi} (1+\sin x+\cos x)\sqrt{\cos^2 x - 2\sin x\cos x + \sin^2 x + 1} dx\). 6. Multiply the result of step 5 by \(2\pi\) to obtain the surface area: \(A = 2\pi I\). Use a computer algebra system, such as Wolfram Alpha, to compute the integral in step 5 and obtain the final surface area.

Step-by-step solution

Find the derivative of the curve with respect to x

To find the first derivative of the curve \(y=1+\sin x+\cos x\) with respect to \(x\), we will differentiate it. The first derivative is given by \(\frac{dy}{dx} = \cos x - \sin x\).

Square the derivative and add 1

Now, square the derivative and add 1: \((\cos x - \sin x)^2 + 1 = \cos^2 x - 2\sin x\cos x + \sin^2 x + 1\)

Find the square root of the result

Next, we need to find the square root of the result from step 2: \(\sqrt{\cos^2 x - 2\sin x\cos x + \sin^2 x + 1}\).

Multiply the curve by the result of step 3

Multiply the curve \(y=1+\sin x+\cos x\) by the result of step 3: \((1+\sin x+\cos x)\sqrt{\cos^2 x - 2\sin x\cos x + \sin^2 x + 1}\).

Integrate the result from step 4 from 0 to \(\pi\)

Now, integrate the expression from step 4 over the interval [0, \(\pi\)]: \(\int_{0}^{\pi} (1+\sin x+\cos x)\sqrt{\cos^2 x - 2\sin x\cos x + \sin^2 x + 1} dx\). At this point, enter the expression into a computer algebra system, such as Wolfram Alpha, to compute the integral.

Multiply the result of step 5 by \(2\pi\) to obtain the surface area

Let the result of the integral in step 5 be denoted by "I". The surface area A is obtained by multiplying I by \(2\pi\): \(A = 2\pi I\). Enter the value of I into the formula to compute the final surface area.

Key concepts

Calculus

Calculus is a branch of mathematics that focuses on the study of change, via derivatives and integrals. It offers powerful tools for understanding and interpreting the behavior of functions. In this particular exercise, we use calculus to find the surface area of a solid of revolution, which is a 3D shape formed by rotating a 2D curve around an axis. This involves several steps where differentiation and integration are key components.

Differentiation helps us to understand how the function changes at any given point, which is the first step in determining the surface area of revolution. We differentiate the curve to find the rate of change along the x-axis. Integration, on the other hand, accumulates these changes over the given interval, thus helping to calculate the total surface area generated. Understanding these fundamental calculus concepts is essential for tackling complex mathematical problems involving surface areas and volumes.

Integration

Integration, in this context, is the method used to calculate the surface area of the solid formed by rotating a curve about an axis. Essentially, it sums up infinitesimally small slices of the surface area to find the total area.

To solve this problem, we first multiply the given function by the square root of the sum of one and the square of its derivative. This represents an infinitesimally small element of the surface. We then integrate this expression over the given interval from 0 to \(\pi\\), which accounts for the full length of the curve's rotation. This integration step consolidates these small contributions into a total surface area before multiplying by \(2\pi\\), due to the rotation around the x-axis.

Such integration practices not only return the surface area but also enrich your understanding of how calculus operates to approximate and exact seemingly complex concepts in geometry.

Parametric Equations

Parametric equations are a powerful tool in calculus that allow us to represent curves in terms of a parameter, often time or another variable. While the curve in the original exercise is given as a function of x, learning about parametric equations can enrich your understanding. They allow the description of more complex curves and surfaces, which might not be easily expressed in terms of y or x alone.

For instance, considering the rotation of the curve as a parametric problem provides deeper insight into potential methods for solving it. Here, both x and y could be expressed in terms of a third variable, such as \( heta\), which would then lend itself to differential calculus techniques customized for parametric forms. These techniques can often simplify integration and differentiation by separating complex interdependencies between x and y.

Computer Algebra Systems

Computer Algebra Systems (CAS) are software programs designed to handle and manipulate mathematical equations symbolically. In this exercise, utilizing a CAS is essential for integrating the complex expression derived earlier. Programs like Wolfram Alpha, Mathematica, or Maple simplify this task, performing symbolic manipulation to compute integrals that may be difficult or cumbersome to evaluate by hand.

Using CAS involves entering the integral expression from the problem, allowing the software to process and return an accurate result. This step helps ensure precision in mathematical computations, which is vital for complex integrations that arise in calculus.

Thus, while understanding the underlying mathematics is crucial, the availability and use of CAS streamline the process, allowing students and professionals to focus on interpreting results and applying them to real-world problems.