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Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter2: Systems Of Linear Equations

Section2.4: Applications

Problem 2EQ: 2. Suppose that in Example 2.27, 400 units of food A, 500 units of B, and 600 units of C are placed...

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Question

Evaluate the definite integral. Use a graphing utility to verify your result.

Expert Solution

Step 1

Apply the properties of integration, to split the given integration into two terms.

$\begin{array}{rcl} \int_{- 1}^{1} (t^{2} - 2) d t & = & \int_{- 1}^{1} (t^{2} d t - 2 d t) \\ = & \int_{- 1}^{1} t^{2} d t - \int_{- 1}^{1} 2 d t \\ = & \int_{- 1}^{1} t^{2} d t - 2 \int_{- 1}^{1} d t \end{array}$

Step 2

Use the integration formula $\int x^{n} d x = \frac{x^{n + 1}}{n + 1} + C$ , to evaluate the given integral.

$\begin{array}{rcl} \int_{- 1}^{1} (t^{2} - 2) d t & = & \int_{- 1}^{1} t^{2} d t - 2 \int_{- 1}^{1} d t \\ = & {[\frac{t^{3}}{3}]}_{- 1}^{1} - 2 {[t]}_{- 1}^{1} \\ = & (\frac{1}{3} - (- \frac{1}{3})) - 2 [1 - (- 1)] \\ = & \frac{1}{3} + \frac{1}{3} - 2 (2) \\ = & \frac{2}{3} - 4 \\ = & - \frac{10}{3} \end{array}$

Therefore, $\begin{array}{rcl} \int \end{array} \begin{array}{rcl} - 1 1 \end{array} \begin{array}{rcl} (t^{2} - 2) \end{array} \begin{array}{rcl} d \end{array} \begin{array}{rcl} t \end{array} \begin{array}{rcl} = \end{array} \begin{array}{rcl} - \end{array} \begin{array}{rcl} \frac{10}{3} \end{array}$ .

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