Chapter 8: Q. 67 (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 .
$\int_{1}^{2} x^{2} \sin (5 x^{2}) d x$

Short Answer

Expert verified

The approximate value is -493.12 .

Step by step solution

Step 1. Given information .

Consider the given integral $\int_{1}^{2} x^{2} \sin (5 x^{2}) d x$ .

Step 2. Using the result of sin x from question 51 - 60 and theorem 7.38 .

The result of $\sin x = \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k}}{(2 k + 1)!} x^{2 k + 1}$ .

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,. Furthermore, the sign of the difference L − Sn is the sign of the coefficient of the term .

Step 3. Find the value .

$\int_{1}^{2} x^{2} \sin (5 x^{2}) d x = \int_{1}^{2} \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k}}{(2 k + 1)!} 5 x^{4 (2 k + 1)} d x = \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k}}{(2 k + 1)!} \int_{1}^{2} 5 x^{8 k + 4} d x = \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k}}{(2 k + 1)!} 5 {[\frac{x^{8 k + 5}}{8 k + 5}]}_{1}^{2} = \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k}}{(2 k + 1)!} [\frac{5 \cdot 2^{8 h + 5}}{8 k + 5}]$

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

$\Rightarrow 32 - 525 \cdot 125 + . . . . . . . \Rightarrow - 493.12$

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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 .
$\int_{1}^{2} x^{2} \sin (5 x^{2}) d x$

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 .

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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 .∫12x2sin5x2dx

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

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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 .
$\int_{1}^{2} x^{2} \sin (5 x^{2}) d x$