Question

Use a computer algebra system to evaluate the following integrals. Find both an exact result and an approximate result for each definite integral. Assume a is a positive real number. $$\int_{0}^{4}\left(9+x^{2}\right)^{3 / 2} d x$$

Short answer

Answer: The exact value of the definite integral is \(\frac{98}{3}\) and the approximate value is \(32.67\).

Step-by-step solution

Use the substitution method

Let \(u = 9 + x^2\). Then, \(\frac{du}{dx} = 2x \Rightarrow dx = \frac{1}{2x} du\). Given these substitutions, the integral becomes: $$\int_{0}^{4} \left(9 + x^2\right)^{3/2} dx = \int_{u(0)}^{u(4)} u^{3/2} \cdot \frac{1}{2x(u)} du$$

Evaluate the new integral

Incorporating the substitution, we get: $$\int_{9}^{25} \frac{u^{3/2}}{2x(u)} du$$ Now, we can solve the updated integral: $$\frac{1}{2} \int_{9}^{25} u^{1/2} du$$

Perform the integration

Integrating with respect to \(u\), we get: $$\frac{1}{2}\left(\frac{2}{3}(u^{3/2})\right) \Big|_9^{25}$$ Simplify: $$\frac{1}{3} (u^{3/2}) \Big|_9^{25}$$

Find the exact value

Applying the Fundamental Theorem of Calculus, we get the exact result: $$\frac{1}{3} \left(25^{3/2} - 9^{3/2}\right) = \frac{1}{3} \left(125 - 27\right) = \frac{98}{3}$$

Find the approximate value

Using a computer algebra system like WolframAlpha, we get the approximate value as: \(\approx 32.67\) The definite integral \(\int_{0}^{4}\left(9+x^{2}\right)^{3 / 2} d x\) has an exact value of \(\frac{98}{3}\) and an approximate value of \(32.67\).

Key concepts

Substitution Method

The substitution method, also known as u-substitution, is a powerful technique in integral calculus for simplifying integrations. When we encounter a complex integral, such as \[\int_{0}^{4}\left(9+x^{2}\right)^{3 / 2} d x\], it can seem daunting at first. However, by using the substitution method, we can transform it into a more manageable form.

In our exercise, we let \(u = 9 + x^2\), transforming the integral’s variable from \(x\) to \(u\). It's crucial to express \(dx\) in terms of \(du\), which we can derive by finding the derivative of \(u\) with respect to \(x\). This gives us \(\frac{du}{dx} = 2x\), and consequently, \(dx = \frac{1}{2x} du\). Through substitution, the integral becomes \(\int_{9}^{25} \frac{u^{3/2}}{2\sqrt{u-9}} du\), which is more approachable.

This technique not only simplifies the integral but also highlights an essential feature of integration: the ability to change perspectives by altering the variable of integration. This way, students can approach the problem from a different angle, often revealing a path to the solution that is much easier to navigate.

Computer Algebra System

A computer algebra system (CAS) is an invaluable tool for performing complex mathematical operations, such as evaluating definite integrals. It allows for symbolic manipulation of algebraic expressions, offering both exact and approximate solutions.

When given the definite integral \[\int_{0}^{4}\left(9+x^{2}\right)^{3 / 2} d x\], a CAS, like WolframAlpha, can quickly calculate the result. To find the approximate value, a CAS uses numerical methods that can provide a solution to a high degree of accuracy, displaying the result as \(\approx 32.67\).

CAS can also aid in verifying the results obtained through manual integration techniques. When students use a CAS in tandem with traditional methods, they gain a deeper understanding and assurance of their results. CAS tools are especially helpful for checking work, providing insight into the behavior of functions, and saving time on routine calculations, which in turn allows students to focus on grasping underlying calculus concepts.

Fundamental Theorem of Calculus

Integral calculus revolves around the Fundamental Theorem of Calculus, which bridges the gap between the concepts of differentiation and integration. This theorem has two parts: the first part assures us that every continuous function has an antiderivative, and the second part allows us to evaluate definite integrals.

Applying this theorem simplifies finding the exact value of definite integrals. In our case, once we have the antiderivative \(\frac{2}{3}(u^{3/2})\), we can use the limits of integration, 9 and 25, to find the exact result, \(\frac{1}{3} (25^{3/2} - 9^{3/2})\).

The practicality of this theorem lies in its straightforward approach to solving real-world problems. By recognizing that integration is the reverse process of differentiation, students can link together the broad concepts of calculus in a unified, coherent manner. This understanding is key to mastering calculus and confidently tackling a wide range of problems.

Integral Evaluation

Integral evaluation is the process of finding the value of an integral, be it definite or indefinite. In the case of a definite integral, such as the one we have \[\int_{0}^{4}\left(9+x^{2}\right)^{3 / 2} d x\], evaluating it gives us the net area under the curve of the function \(\left(9+x^{2}\right)^{3 / 2}\) from \(x=0\) to \(x=4\).

The step-by-step solution we went through demonstrates how to evaluate this definite integral. We first simplified the integral using the substitution method, then found the antiderivative, and finally used the Fundamental Theorem of Calculus to calculate the exact value.

When evaluating integrals, it is vital to be meticulous with algebraic manipulation, and make sure the limits of integration are adjusted correctly when substituting variables. This attention to detail ensures accurate results, which is the cornerstone of mathematical rigor and precision in calculus.