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. $$\int_{0}^{4}\left(9+x^{2}\right)^{3 / 2} d x$$

Short answer

Question: Solve the definite integral $\int_{0}^{4}\left(9+x^{2}\right)^{3 / 2} d x$ and compare the exact and approximate values using a computer algebra system. Answer: The exact value of the integral is $\frac{1}{5} (25^{5/2} - 9^{5/2})$. The approximate value can be found using a numerical method like Simpson's Rule or any other available method in a CAS. When comparing the exact and approximate values, they should be very close depending on the numerical method and accuracy used in the CAS.

Step-by-step solution

Setup of the integral substitution method

We will perform a substitution method to find the exact value of the integral. Let's make a substitution as follows: $$u=9+x^2$$ Then we have: $$du = 2x dx$$ Now, we need to change the limits of the integration according to the new variable "u". When \(x=0\), we get \(u=9\). When \(x=4\), we get \(u=25\).

Perform substitution in the integral

Now let's perform the substitution in our 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 will solve the integral according to the new variable "u".

Solve the integral for the variable "u"

The integral becomes: $$\frac{1}{2} \int_{9}^{25} u^{3/2}du$$ Now, the antiderivative of \(u^{3/2}\) is \(\frac{2}{5}u^{5/2}\). Applying the antiderivative to the integral, we have: $$\frac{1}{2} \left[\frac{2}{5}u^{5/2}\right]_{9}^{25}$$ Now we will evaluate this expression at the upper and lower limits of the integration, and subtract the results to get the final answer.

Evaluate the expression at the limits and obtain the exact value

We first evaluate the expression at the upper limit \(u = 25\) and then at the lower limit \(u = 9\) and subtract the both to get the final answer: $$\frac{1}{2} \left(\frac{2}{5} 25^{5/2} - \frac{2}{5} 9^{5/2}\right) = \frac{1}{5} (25^{5/2} - 9^{5/2})$$ This is the exact value of the integral.

Calculate the approximate value using a numerical method

To find the approximate value of the integral, we will use a numerical method like Simpson's Rule or any other available method in a computer algebra system (CAS). For example, using Simpson's Rule or a numerical integration function in a CAS like Wolfram Alpha (you can use other CAS software for this task) and entering the original function, you will obtain an approximate value.

Compare the exact and approximate values

At this point, you should have the exact and approximate values for the given integral. Now, you can compare the exact value obtained in Step 4 with the approximate value found through a numerical method in Step 5. Notice that the values may not be the same, but they should be very close depending on the numerical method and accuracy used in the CAS.