Problem 53 - Calculus Early Transcendentals

Question

Use a computer algebra system to evaluate the following indefinite integrals. Assume that a is a positive real number. $$\int\left(a^{2}-x^{2}\right)^{3 / 2} d x$$

Step-by-Step Solution

Answer

Question: Evaluate the indefinite integral of \((a^2 - x^2)^{3/2}\) using a computer algebra system, where \(a\) is a positive real number. Answer: The indefinite integral of \((a^2 - x^2)^{3/2}\) is given by: $$\int\left(a^{2}-x^{2}\right)^{3 / 2} dx = \frac{2}{15} \cdot a^2 \cdot (a^2 - x^2)^\frac{1}{2} \cdot (8x^2 - 3a^2) + C$$

Step 1: Set up the problem in a CAS

Choose a computer algebra system such as Wolfram Alpha, Symbolab, or SageMath. Input the integral in the correct syntax. For example, in Wolfram Alpha, input "integrate (a^2 - x^2)^(3/2) dx" (without the quotes). Note that the syntax for other CAS might be different.

Step 2: Evaluate the integral

Run the CAS with the input to find the integral. Most CAS will output the result in terms of \(x\) and \(a.\) The result will be the antiderivative of the integrand. For this exercise, the indefinite integral is: $$\frac{2}{15} \cdot a^2 \cdot (a^2 - x^2)^\frac{1}{2} \cdot (8x^2 - 3a^2) + C$$

Step 3: Add the constant of integration

The final step in finding the indefinite integral is to add the constant of integration, which is denoted as \(C.\) In this case, the final result of the integral is: $$\int\left(a^{2}-x^{2}\right)^{3 / 2} dx = \frac{2}{15} \cdot a^2 \cdot (a^2 - x^2)^\frac{1}{2} \cdot (8x^2 - 3a^2) + C$$

Key Concepts

Computer Algebra Systems

Computer Algebra Systems (CAS) are powerful tools designed to solve mathematical problems, compute derivatives, integrals, and much more. For students tackling complex problems, especially indefinite integrals, a CAS can be incredibly useful. To use a CAS effectively, it's essential to input the problem using the correct syntax. For instance, to evaluate \(\int (a^2 - x^2)^{\frac{3}{2}} dx\), it's vital to express the integral in a way the computer system can interpret.

Once entered, the CAS processes the information using advanced algorithms to find the antiderivative. It's almost like having a digital math assistant at your disposal, saving time and ensuring accuracy. However, remember that the reliance on technology should not replace the fundamental understanding of the underlying mathematics. This approach is best used as a supplement to your studies or a means to check your work.

Antiderivative

The antiderivative, also known as the indefinite integral, is essentially the reverse process of taking a derivative. It plays a crucial role in calculus and is denoted by the integral sign (\(\int\)). Finding the antiderivative of a function involves determining the original function whose derivative gives the integrand.

In our example \(\int(a^2 - x^2)^{\frac{3}{2}} dx\), we seek the function that, once differentiated, would equal \(a^2 - x^2)^{\frac{3}{2}}\). The process often involves recognizing patterns that align with known derivatives and applying integration rules. Mastery of antiderivatives is not only academically rewarding but also practical as it forms the basis for solving complex real-world problems involving rates of change.

Constant of Integration

In the final stage of finding an indefinite integral, introducing the constant of integration is a critical step. Because indefinite integrals represent a family of functions, the constant of integration \(C\) encompasses all possible constant values that could be added to the antiderivative.

This concept is like a '+C' after solving for an indefinite integral—such as \(\int(a^2 - x^2)^{\frac{3}{2}} dx\)—to acknowledge that there are multiple functions that could differentiate to give the same original function plus some constant. Each of these functions is correct, and without additional information such as an initial condition or boundary value, \(C\) cannot be determined and hence must be included in the final answer.