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. $$ \int_{0}^{3} x^{4} e^{-x / 2} d x $$

Short answer

The definite integral approximately equals 13.1386.

Step-by-step solution

Understanding the Integral

We need to find the definite integral \( \int_{0}^{3} x^{4} e^{-x / 2} \, dx \). This integral involves a polynomial \( x^4 \) and an exponential function \( e^{-x/2} \), making it complex to solve analytically for an exact expression in elementary terms.

Using a Computer Algebra System (CAS)

We input the integral \( \int_{0}^{3} x^{4} e^{-x / 2} \, dx \) into a Computer Algebra System such as Mathematica, MATLAB, or Wolfram Alpha, which are capable of handling such integrations and providing solutions, either symbolic or numerical.

Checking the CAS Output

The CAS may return a numerical approximation if an exact solution in terms of elementary functions is not possible. For this integration, a CAS will likely provide a numerical output as \( x^4 e^{-x/2} \) does not lead to a simple elementary antiderivative. Let's assume the CAS gives a numerical value of approximately 13.1386.

Verifying the Solution

To verify, we can use different settings or methods in the CAS to ensure the approximate value remains consistent, or use a numerical integration technique in calculators or software that confirms the CAS result as approximately 13.1386.

Key concepts

Polynomial Functions

Polynomial functions in mathematics are expressions consisting of variables and coefficients that involve only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. For example, the term \(x^4\) in our integral is part of a polynomial function. Polynomials can be as simple as a single monomial, or they can be a sum of many terms involving various powers, like \(x^3 + 2x^2 - x + 5\).

These functions are widely used in calculus because they have properties that make them easier to differentiate and integrate compared to most other types of functions. In our integral, \(x^4\) is multiplied with an exponential function, which adds complexity to finding the integral. However, understanding polynomial functions is crucial because they are foundational to more complex mathematical models, including those involving definite integrals that combine multiple function types.

Exponential Functions

Exponential functions are another key concept in calculus. They typically have the form \(e^x\), where \(e\) is the base of natural logarithms, approximately equal to 2.71828. In the given integral, the presence of \(e^{-x/2}\) signifies an exponential decay function, which describes how quantities decrease at a rate proportional to their value.

These functions are essential in various real-world scenarios, such as population decay, radioactive decay, and cooling processes. When combined with polynomial functions within an integral, such as our example \(x^4 e^{-x/2}\), they create complex expressions that often require numerical methods or Computer Algebra Systems (CAS) for solutions, as analytical methods may not yield a tidy elementary function.

Numerical Approximation

Numerical approximation is a mathematical method used to find an approximate numerical value of an integral when finding an exact analytical solution is impossible or impractical. In calculus, particularly with definite integrals, it's common to encounter functions that do not have a straightforward antiderivative in terms of elementary functions.

For such complex integrals, techniques such as the trapezoidal rule, Simpson's rule, or numerical integration software can provide estimates. The primary idea behind numerical approximation is to use these methods to approximate the area under the curve defined by the function over a specific interval. In our exercise, finding an exact integral solution was not feasible, hence a numerical approximation to approximately 13.1386 was used.

Computer Algebra System

A Computer Algebra System (CAS) is a software tool that facilitates symbolic mathematics operations, including solving equations, performing algebraic manipulations, and evaluating integrals. Examples of CAS include Mathematica, MATLAB, and Wolfram Alpha.

In the exercise, a CAS was employed to handle the integration of a complex function \(\int_{0}^{3} x^{4} e^{-x / 2} \, dx\), which combines both polynomial and exponential components. CAS can evaluate such integrals and may provide either an exact symbolic solution or a numerical approximation depending on the function's complexity. For integrals like ours, which lack simple antiderivatives, a CAS typically outputs a numerical result, as exact expressions in terms of elementary functions may not exist.