Chapter 8: Q. 63 (page 702)

Use the results from Exercises 51–60 and Theorem 7.38 to approximate the values of the definite integrals in Exercises 61–70 to within 0.001 of their values.
$\int_{- 1}^{1} e^{- x^{2} / 3} d x$

Short Answer

Expert verified

The approximate value is $- 5 \cdot 978$ .

Step by step solution

Step 1. Given information .

Consider the given integral $\int_{- 1}^{1} e^{- x^{2} / 3} d x$ .

Step 2. Using the result of sin x from question 51 - 60 and theorem 7·38 .

The result of $e^{x} = \sum_{k = 1}^{\infty} \frac{1}{k!} x^{k}$ .

Theorem 7.38 - Let L be the sum of an alternating series satisfying the

hypotheses of the alternating series test. For any term Sn in the sequence of partial sums, $|L - S_{n}| < a_{n + 1}$ . Furthermore, the sign of the difference L − Sn is the sign of the coefficient of the term ..

Step 3. Find the value .

$\int_{- 1}^{1} e^{- x^{2} / 3} d x = \int_{- 1}^{1} \sum_{k = 0}^{\infty} \frac{{(- x^{2} / 3)}^{k}}{k!} = \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k}}{3 k!} \int_{- 1}^{1} (x^{2 k}) = \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k}}{3 k!} {[\frac{x^{2 k + 1}}{2 k + 1}]}_{- 1}^{1} = \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k}}{3 k!} (\frac{2}{2 k + 1})$

Substitute $k = 0, 1, 2, 3, . . . . . . . . . . . \infty$

$\Rightarrow 2 - \frac{1}{9} + \frac{1}{180} - \frac{1}{1270080} \Rightarrow 2 - 0 \cdot 111 + 0 \cdot 00555 - 7 \cdot 873 \Rightarrow - 5 \cdot 978$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Use the results from Exercises 51–60 and Theorem 7.38 to approximate the values of the definite integrals in Exercises 61–70 to within 0.001 of their values.
$\int_{- 1}^{1} e^{- x^{2} / 3} d x$

Short Answer

Step by step solution

Step 1. Given information .

Step 2. Using the result of sin x from question 51 - 60 and theorem 7·38 .

Step 3. Find the value .

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Geometry

Logic and Functions

Theoretical and Mathematical Physics

Statistics

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Use the results from Exercises 51–60 and Theorem 7.38 to approximate the values of the definite integrals in Exercises 61–70 to within 0.001 of their values.∫-11e-x2/3dx

Short Answer

Step by step solution

Step 1. Given information .

Step 2. Using the result of sin x from question 51 - 60 and theorem 7·38 .

Step 3. Find the value .

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Geometry

Logic and Functions

Theoretical and Mathematical Physics

Statistics

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Use the results from Exercises 51–60 and Theorem 7.38 to approximate the values of the definite integrals in Exercises 61–70 to within 0.001 of their values.
$\int_{- 1}^{1} e^{- x^{2} / 3} d x$