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Chapter 8: Q. 67 (page 702)

Use the results from Exercises 51鈥�60 and Theorem 7.38 to approximate the values of the definite integrals in Exercises 61鈥�70 to within 0.001 of their values .
${鈭�}_{1}^{2} x^{2} \sin (5 x^{2}) d x$

Short Answer

Expert verified

The approximate value is -493.12 .

Step by step solution

01

Step 1. Given information .

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

02

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

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

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 鈭� Sn is the sign of the coefficient of the term .

03

Step 3. Find the value .

${鈭�}_{1}^{2} x^{2} \sin (5 x^{2}) d x = {鈭�}_{1}^{2} {鈭�}_{k = 0}^{鈭�} \frac{{(- 1)}^{k}}{(2 k + 1)!} 5 x^{4 (2 k + 1)} d x = {鈭�}_{k = 0}^{鈭�} \frac{{(- 1)}^{k}}{(2 k + 1)!} {鈭�}_{1}^{2} 5 x^{8 k + 4} d x = {鈭�}_{k = 0}^{鈭�} \frac{{(- 1)}^{k}}{(2 k + 1)!} 5 {[\frac{x^{8 k + 5}}{8 k + 5}]}_{1}^{2} = {鈭�}_{k = 0}^{鈭�} \frac{{(- 1)}^{k}}{(2 k + 1)!} [\frac{5 路 2^{8 h + 5}}{8 k + 5}]$

Substitute $k = 0, 1, 2, 3 . . . . . . . . . . . . . . . . . 鈭�$

$鈬� 32 - 525 路 125 + . . . . . . . 鈬� - 493.12$

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Most popular questions from this chapter

Let ${鈭�}_{k = 0}^{鈭�} 鈥� a_{k} x^{k}$ be a power series in $x$ with a finite radius of convergence $p$ . Prove that if the series converges absolutely at either $卤 p$ , then the series converges absolutely at the other value as well.

What is meant by the interval of convergence for a power series in $x - x_{0}$ ? How is the interval of convergence determined? If a power series in $x - x_{0}$ has a nontrivial interval of convergence, what types of intervals are possible.

The second-order differential equation

$x^{2} y^{''} + x y^{'} + (x^{2} - p^{2}) = 0$

where p is a non-negative integer, arises in many applications in physics and engineering, including one model for the vibration of a beaten drum. The solution of the differential equation is called the Bessel function of order p, denoted by $J_{p} (x)$ . It may be shown that $J_{p} (x)$ is given by the following power series in x :

$J_{p} (x) = {鈭�}_{k = 0}^{鈭�} \frac{(- 1)^{k}}{k! (k + p)! 2^{2 k + p}} x^{2 k + p}$

What is the interval of convergence for $J_{1} (x)$ ?

If $f (x) = 4 x^{3} - 5 x^{2} + 6 x + 1 a n d P_{3} (x)$ is the third Taylor polynomial for f at 鈭�1, what is the third remainder $R_{3} (x)$ ? What is $R_{4} (x)$ ? (Hint: You can answer this question without finding any derivatives.)

What is meant by the interval of convergence for a power series in $x$ ? How is the interval of convergence determined? If a power series in $x$ has a nontrivial interval of convergence, what types of intervals are possible?

See all solutions

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