Problem 55 - Applied Calculus

Question

For each definite integral: a. Evaluate it "by hand." b. Check your answer by using a graphing calculator. $$ \int_{0}^{4} \sqrt{x^{2}+9} x d x $$

Step-by-Step Solution

Answer

The evaluated integral is 32.67.

Step 1: Choose the Method of Integration

The function \( \sqrt{x^2+9} x \) in the integral can be integrated using substitution. Let's apply the substitution method.

Step 2: Perform Substitution

Let \( u = x^2 + 9 \). Then, differentiate \( u \) to get \( du = 2x \, dx \). Thus, \( x \, dx = \frac{1}{2} du \). Change the limits: when \( x = 0 \), \( u = 9 \); when \( x = 4 \), \( u = 25 \).

Step 3: Rewrite the Integral with Substitution

The integral becomes \[ \int_{9}^{25} \sqrt{u} \cdot \frac{1}{2} du \]. Simplify this to \[ \frac{1}{2} \int_{9}^{25} u^{1/2} du \].

Step 4: Integrate the Substituted Integral

Use the power rule for integration \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \). Apply this rule to get \[\frac{1}{2} \left[ \frac{u^{3/2}}{3/2} \right]_{9}^{25} = \frac{1}{3} \left[ u^{3/2} \right]_{9}^{25} \\].

Step 5: Evaluate between the Limits

Calculate \[ \frac{1}{3} \left( 25^{3/2} - 9^{3/2} \right) \]. Find \( 25^{3/2} = 125 \) and \( 9^{3/2} = 27 \). Thus, the evaluated integral is \[ \frac{1}{3} (125 - 27) = \frac{1}{3} \times 98 = 32.67 \].

Step 6: Verify with Graphing Calculator

Use a graphing calculator to compute the definite integral \( \int_{0}^{4} \sqrt{x^{2}+9} x \, dx \). Confirm that the result matches with \( 32.67 \).

Key Concepts

Substitution Method

The substitution method is a powerful technique used to simplify difficult integrals by transforming them into a more manageable form. This process involves replacing a part of the integrand with a new variable, usually denoted as \( u \), to make the integral easier to solve. In the given problem, we deal with the function \( \sqrt{x^2+9}x \).

To use substitution, we first set \( u = x^2 + 9 \). This choice simplifies the radical expression inside the integral. We then differentiate \( u \) to get \( du = 2x \, dx \), and rearrange it to find that \( x \, dx = \frac{1}{2} du \). This step is crucial because it allows us to express the original integral in terms of \( u \) and \( du \).

• The original limits of integration in terms of \( x \) (0 to 4) are converted to limits in terms of \( u \) (9 to 25) by substituting the values into \( u = x^2 + 9 \).
• The integral now reads \( \frac{1}{2} \int_{9}^{25} u^{1/2} \, du \).
• This substitution has simplified the process by eliminating the product of functions under the radical.

Using the substitution method requires careful choice of substitution and adjustment of integration limits. It's a method that transforms the integral, making it more straightforward to evaluate.

Power Rule for Integration

The power rule for integration is a fundamental concept in calculus used to solve integrals of expressions that include powers of a variable. Specifically, the power rule states that \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \), where \( n eq -1 \). This rule provides an easy way to find antiderivatives for most simple power functions.

In the transformed integral \( \frac{1}{2} \int_{9}^{25} u^{1/2} \, du \), we identify the power as \( n = \frac{1}{2} \). The integration using the power rule follows these steps:

• We calculate \( \int u^{1/2} \, du = \frac{u^{3/2}}{3/2} = \frac{2}{3} u^{3/2} \).
• So, the expression becomes \( \frac{1}{2} \cdot \frac{2}{3} u^{3/2} = \frac{1}{3} u^{3/2} \).
• We then evaluate it from 9 to 25: \( \frac{1}{3} [u^{3/2}]_{9}^{25} \).

Using the power rule simplifies the calculation, making it efficient to find the definite integral's value without complex arithmetic. Understanding this rule is essential for evaluating integrals where functions have exponentiated terms.

Graphing Calculator Verification

Verifying solutions using a graphing calculator serves as a valuable step in ensuring the accuracy of manual calculations. After computing the definite integral manually, it's prudent to check your result using technology. This process helps catch errors that might have occurred during hand calculations and confirms the correctness of your solution.

For this problem, after finding the integral \( \int_{0}^{4} \sqrt{x^2+9} x \, dx = 32.67 \) using the substitution and power rule methods, we proceed to enter this integral into a graphing calculator:

• Set the integral function in the calculator: \( \int_{0}^{4} \sqrt{x^2+9} x \, dx \).
• Ensure all equations and limits are entered correctly.
• The calculator performs the computation, ideally outputting the same result: \( 32.67 \).

This verification step gives confidence that the manual calculations align with technological evaluations. While graphing calculators are powerful tools, users should also be comfortable with verifying and cross-verifying results when tackling math problems.