Problem 35 - Calculus Early Transcendentals: Pearson New International Edition

Question

Use a CAS to evaluate the definite integrals. If the CAS does not give an exact answer in terms of elementary functions, then give a numerical approximation. $$ \int_{1}^{4} \frac{\sqrt{t}}{1+t^{8}} d t $$

Step-by-Step Solution

Answer

The numerical approximation of the integral is approximately 0.868.

Step 1: Understand the Problem

We need to evaluate the definite integral \( \int_{1}^{4} \frac{\sqrt{t}}{1+t^{8}} \, dt \). A CAS (Computer Algebra System) will be used, as this integral might not have a solution in terms of elementary functions.

Step 2: Use CAS to Evaluate the Integral

Input the integral \( \int_{1}^{4} \frac{\sqrt{t}}{1+t^{8}} \, dt \) into a CAS. The CAS software will automatically determine if an exact answer in terms of elementary functions is possible.

Step 3: Interpret CAS Output

The CAS does not provide an exact answer in terms of elementary functions for this integral. Instead, it gives a numerical approximation of the integral value. The numerical approximation given by the CAS is approximately 0.868.

Key Concepts

Numerical Approximation

When evaluating definite integrals, sometimes an exact solution using traditional analytical methods might not be possible, especially when it involves complex expressions. In such cases, numerical approximation methods become essential. A numerical approximation provides a step-by-step procedure to evaluate an integral to a certain degree of accuracy rather than attempting to find an exact solution. This is especially useful for integrals that involve functions like the one in our exercise: \[ \int_{1}^{4} \frac{\sqrt{t}}{1+t^{8}} \, dt \]The given problem asks to use a Computer Algebra System (CAS) to evaluate the integral. CAS systems often employ numerical techniques like the Trapezoidal Rule, Simpson’s Rule, or more advanced algorithms to estimate a value. While these approximations might not be "exact," they are generally close enough for practical purposes, such as scientific or engineering calculations. In this exercise, the CAS software provides a numerical approximation of approximately 0.868, offering a practical, usable solution when an exact answer isn't feasible.

Elementary Functions

Elementary functions are the building blocks of calculus that include algebraic functions, trigonometric functions, exponential functions, and logarithms. They form the backbone of calculus operations, providing a framework through which we can understand and evaluate more complex functions. In calculus, an integral that resolves to elementary functions can often be addressed using antiderivative techniques, where you find a function whose derivative matches the integrand. However, the given integral:\[ \int_{1}^{4} \frac{\sqrt{t}}{1+t^{8}} \, dt \]involves a composition that does not readily simplify into elementary functions. The presence of higher powers, such as \( t^{8} \), combined with square roots, indicates complex behavior. When a CAS evaluates such integrals, it attempts to express them in terms of elementary functions, but sometimes, as in this case, it finds that no such expression can exist, thus requiring a numerical approximation.

Computer Algebra System (CAS)

A Computer Algebra System (CAS) is a software tool that performs symbolic mathematics, providing capabilities beyond simple numeric calculations. These systems are beneficial for complex mathematics involving algebra, calculus, and other domains where symbolic manipulation is required.In the context of our exercise, the CAS attempts to evaluate the integrals and outputs either in terms of known mathematical objects (like elementary functions) or via numerical approximations when exact solutions are impossible or extremely complicated. The process typically involves:

• Input: Users input the integral expression, such as \( \int_{1}^{4} \frac{\sqrt{t}}{1+t^{8}} \, dt \).
• Processing: The CAS employs algorithms to attempt to find a solution.
• Output: The CAS returns either an analytical solution if possible, or a numerical approximation.

CAS systems are incredibly powerful as they can handle a vast range of mathematical problems, providing students and professionals with a tool that aids in understanding complex computations without manual labor. In our case, the system was used to produce an approximation for an integral that is not easily expressed in conventional terms.