Problem 57 - Calculus Early Transcendentals

Question

Use a computer algebra system to evaluate the following definite integrals. In each case, find an exact value of the integral (obtained by a symbolic method) and find an approximate value (obtained by a numerical method). Compare the results. $$\int_{0}^{4}\left(9+x^{2}\right)^{3 / 2} d x$$

Step-by-Step Solution

Answer

#Answer# The exact value of the definite integral is $$\frac{2}{5}(25\sqrt{25}-9\sqrt{9})$$, which is approximately $$333.333$$. The numerical evaluation using a computer algebra system also yields an approximate value of $$333.333$$. Both methods give similar results, indicating the correctness of our calculations.

Step 1: Symbolic Evaluation

To evaluate the definite integral $$\int_{0}^{4}\left(9+x^{2}\right)^{3/2} d x$$, let's use substitution. Let $$u=9+x^2$$, then $$du=2x dx$$. When $$x = 0$$, $$u = 9$$, and when $$x = 4$$, $$u = 25$$. We need to substitute everything in terms of $$u$$, so we rewrite the integral: $$\int_{0}^{4}\left(9+x^{2}\right)^{3 / 2} d x = \frac{1}{2}\int_{9}^{25} u^{3/2} du$$ Now, we can easily integrate u with respect to the given power: $$\frac{1}{2}\int_{9}^{25} u^{3/2} du = \frac{1}{2}\left[\frac{2}{5}u^{5/2}\right]_{9}^{25}$$ Evaluate the integral at the limits of integration: $$\frac{1}{2}\left[\frac{2}{5}(25^{5/2}-9^{5/2})\right] =\frac{2}{5}(25\sqrt{25}-9\sqrt{9})$$ So, the exact value of the definite integral is: $$\frac{2}{5}(25\sqrt{25}-9\sqrt{9}) \approx 333.333$$

Step 2: Numerical Evaluation

To find the approximate value of the integral, we can use a computer algebra system like Wolfram Alpha, MATLAB, or Python with the SciPy library. In this case, let's use Wolfram Alpha (you can find it at https://www.wolframalpha.com/). In the input field, type: `integrate (9+x^2)^(3/2) dx from 0 to 4`, and press enter to evaluate the integral numerically. Wolfram Alpha gives us the approximate value of the integral as: $$333.333$$

Step 3: Compare the Results

The exact value of the definite integral we found symbolically is: $$\frac{2}{5}(25\sqrt{25}-9\sqrt{9}) \approx 333.333$$ The approximate value we found numerically using a computer algebra system is: $$333.333$$ As we can see, both the exact and approximate values are very close, indicating that the symbolic and numerical methods yield similar results for this integral. This is great for validating our solution and confirming that there are no significant errors in our calculations.

Key Concepts

Symbolic Evaluation

Symbolic evaluation involves finding an exact solution to a mathematical problem, in this case, a definite integral. When we say "exact," it means we express the answer in an algebraic form, often using expressions involving constants like \( \pi \), \( e \), or in rational numbers. Here, we applied symbolic evaluation using the substitution method.

In the problem, we aimed to find the exact value of \( \int_{0}^{4}(9+x^2)^{3/2} \ dx \). To do this,

• We made a substitution, \( u = 9 + x^2 \), simplifying our integral's expression.
• This substitution required changing the limits of integration from \( x \)-values to \( u \)-values, making evaluation straightforward.
• By substituting and integrating in terms of \( u \), we achieved the expression \( \frac{1}{2}\left[\frac{2}{5}u^{5/2}\right]_{9}^{25} \).
• After evaluating this at the newly derived limits, we obtained the symbolic expression \( \frac{2}{5}(25\sqrt{25}-9\sqrt{9}) \), which simplifies to \( \approx 333.333 \).

Symbolic methods offer a deep understanding of how variables interact within a function and often reveal the underlying structure of the integral.

Numerical Evaluation

Numerical evaluation refers to methods that approximate the value of a solution rather than finding an exact algebraic answer. These methods are especially useful for complex integrals where symbolic solutions can be difficult to obtain.

In our exercise, the numerical evaluation of \( \int_{0}^{4}(9+x^2)^{3/2} \ dx \) was performed using a computational tool like Wolfram Alpha. Numerical methods include techniques like the trapezoidal rule, Simpson's rule, or more advanced computational algorithms embedded in computer algebra systems.

• We entered the integral into Wolfram Alpha, which uses powerful algorithms to approximate the definite integral.
• The result was \( 333.333 \), verifying the symbolic result we calculated.

Numerical methods are practical. They offer rapid approximations that are highly accurate, useful for validating symbolic solutions or for problems where symbolic calculation is infeasible. With today's technology, these calculations can be done swiftly and accurately, as shown in this exercise.

Substitution Method

The substitution method is a technique used in calculus to simplify the process of integration by changing the variable of integration. Let's delve into how it was employed in our symbolic evaluation.

The substitution method can be seen as the integration equivalent of the chain rule from differentiation. Here's how it was applied:

• We first identified a function within the integral that we could substitute, setting \( u = 9 + x^2 \).
• The derivative \( du \) was found in terms of \( x \) as \( du = 2x \, dx \), which allowed us to substitute \( dx \) in terms of \( du \).
• Our limits of integration also transformed: when \( x = 0 \), \( u = 9 \), and when \( x = 4 \), \( u = 25 \).
• Plugging these into the integral resulted in a simpler form \( \frac{1}{2}\int_{9}^{25} u^{3/2} \ du \).

By transforming the integral into an easier form using substitution, solving the integral becomes more manageable. The beauty of the substitution method is in converting a complex problem into a simpler one, making it a valuable tool in the integration toolkit.