Q. 48 Approximate each definite integr... [FREE SOLUTION]

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Chapter 5: Q. 48 (page 492)

Approximate each definite integral in Exercises 41鈥�52 with the indicated method and to the given degree of accuracy. Then calculate the definite integral exactly, and verify that the error bounds you found are accurate.
${鈭�}_{1}^{4} x^{\frac{2}{3}} d x midpoint sum within 0.005$

Short Answer

Expert verified

The required value is $5.449$

Step by step solution

Step 1. Given Information

The given integral is ${鈭�}_{1}^{4} x^{\frac{2}{3}} d x midpoint sum within 0.005$

Step 2. Explanation

The error bound on midpoint sum approximation is given by inequality,

$|E_{M I D (n)}| 鈮� \frac{M {(x_{n} - x_{0})}^{3}}{12 n^{2}} with f'' (x) 鈮� M$

Determine the second order derivative of the given function to have the value of M,

$f' (x) = \frac{2}{3} x^{- \frac{1}{3}} f'' (x) = - \frac{2}{9} x^{- \frac{4}{3}}$

Thus, the value of M that is to be considered for determining the accuracy of midpoint sum is $\frac{2}{9}$ .

Now, determine the number of division for the given function and approximations.

$\frac{2 {(4 - 1)}^{3}}{9 脳 24 {(n)}^{2}} < 0.005 \frac{2 (27)}{216 (n^{2})} < 0.005 n^{2} > \frac{54}{1.08} n > 7.07$

Hence, number of divisions required is 8.

Step 3. Calculation

Divide the given interval into 8 subparts to determine the width of each interval.

$鈭� x = \frac{x_{8} - x_{0}}{8} = \frac{4 - 1}{8} = \frac{3}{8}$

Determine the points of intervals, that is $1, \frac{11}{8}, \frac{14}{8}, \frac{17}{8}, \frac{20}{8}, \frac{23}{8}, \frac{26}{8}, \frac{29}{8}, 4$

Create a table of values for the function at the midpoint of intervals.

Point of interval	function
$1$	$1.121$
$\frac{11}{8}$	$1.347$
$\frac{14}{8}$	$1.554$
$\frac{17}{8}$	$1.749$
$\frac{20}{8}$	$1.933$
$\frac{23}{8}$	$2.109$
$\frac{26}{8}$	$2.278$
$\frac{29}{8}$	$2.440$

Step 4. Calculation

Use these values to determine the midpoint sum approximation.

$M I D (8) = \frac{3}{8} (f (\frac{x_{0} + x_{1}}{2}) + f (\frac{x_{1} + x_{2}}{2}) + f (\frac{x_{2} + x_{3}}{2}) + f (\frac{x_{3} + x_{4}}{2}) + f (\frac{x_{4} + x_{5}}{2}) + f (\frac{x_{5} + x_{6}}{2}) + f (\frac{x_{6} + x_{7}}{2}) + f (\frac{x_{7} + x_{8}}{2})) = \frac{3}{8} (1.121 + 1.347 + 1.554 + 1.749 + 1.933 + 2.109 + 2.278 + 2.440) = \frac{3}{8} (14.531) = 5.449$

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Approximate each definite integral in Exercises 41鈥�52 with the indicated method and to the given degree of accuracy. Then calculate the definite integral exactly, and verify that the error bounds you found are accurate.
${鈭�}_{1}^{4} x^{\frac{2}{3}} d x midpoint sum within 0.005$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Explanation

Step 3. Calculation

Step 4. Calculation

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Approximate each definite integral in Exercises 41鈥�52 with the indicated method and to the given degree of accuracy. Then calculate the definite integral exactly, and verify that the error bounds you found are accurate. 鈭�14x23dxmidpointsumwithin0.005

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Explanation

Step 3. Calculation

Step 4. Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Statistics

Geometry

Decision Maths

Logic and Functions

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Approximate each definite integral in Exercises 41鈥�52 with the indicated method and to the given degree of accuracy. Then calculate the definite integral exactly, and verify that the error bounds you found are accurate.
${鈭�}_{1}^{4} x^{\frac{2}{3}} d x midpoint sum within 0.005$