Chapter 8: Q. 68 (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.
$\int_{0.5}^{2} x^{3} \cos (\frac{x}{2}) dx$

Short Answer

Expert verified

$\int_{0.5}^{2} x^{3} \cos (\frac{x}{2}) dx \approx - 0.342$

Step by step solution

Step 1. Given information is:

$\int_{0.5}^{2} x^{3} \cos (\frac{x}{2}) dx$

Step 2. Approximating the values

$From Q 58 . \int_{0.5}^{2} x^{3} \cos (\frac{x}{2}) dx = \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k}}{(2 k)!} {(\frac{1}{2})}^{2 k + 3} [\frac{2^{2 k + 4} - {(0.5)}^{2 k + 4}}{2 k + 4}] This implies that to approximate the values of integrals, put the value of k \int_{0.5}^{2} x^{3} \cos (\frac{x}{2}) dx = \frac{1}{2^{3} \cdot 4} (2^{4} - 0 . 5^{4}) - \frac{1}{2! \cdot 2^{5} \cdot 6} (2^{6} - 0 . 5^{6}) + \frac{1}{4! \cdot 2^{7} \cdot 8} (2^{8} - 0 . 5^{8}) = 0.4980 - 0.1666 + 0.0104 - 0.000277 + . . . \approx - 0.342$

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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.
$\int_{0.5}^{2} x^{3} \cos (\frac{x}{2}) dx$

Short Answer

Step by step solution

Step 1. Given information is:

Step 2. Approximating the values

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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.∫0.52x3cosx2dx

Short Answer

Step by step solution

Step 1. Given information is:

Step 2. Approximating the values

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Most popular questions from this chapter

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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.
$\int_{0.5}^{2} x^{3} \cos (\frac{x}{2}) dx$