Chapter 5: Q 30 (page 491)

Calculate each definite integral approximation in Exercises 23–40, and then find an error bound for your approximation. If it is possible to calculate the definite integral exactly, then do so and verify that the error bounds you found are accurate
$\int_{1}^{7} \ln (x) d x, t r a p e z o i d s u m, n = 12$

Short Answer

Expert verified

Definite integral is 7.6213

Error <=0.125

Step by step solution

Given Information

$\int_{1}^{7} \ln (x) d x, t r a p e z o i d s u m, n = 12$

Definite integral

$\int_{1}^{7} \ln (x) d x {[x \ln x - x]}_{1}^{7} [7 \ln 7 - 7] - [1 \ln 1 - 1] t h e v a l u e o f d e f i n t e i n t e g r a l i s 7.6213$

Trapezoidal sum

$\int_{a}^{b} f (x) d x = \frac{∆ x}{2} [f (x_0) + 2 f (x_{1}) + 2 f (x_{2}) . . . . . + f (x_{n})] ∆ x = \frac{b - a}{n} = \frac{7 - 1}{12} = \frac{1}{4} l e f t e n d p o i n t o f I n t e r v a l s a r e 1, \frac{3}{2} , 2, \frac{5}{2} , 3, \frac{7}{2} , 4, \frac{9}{2} , 5, \frac{11}{2} , 6, \frac{13}{2} , 7 f (x) = \ln (x) S u b s t i t u t e e a c h e n d p o i n t a n d f i n d o t u t h e s u m s u m = \frac{1}{4} [\ln (1) + 2 \ln (\frac{3}{2}) + 2 \ln (2) + 2 \ln (\frac{5}{2}) + 2 \ln (3) + 2 \ln (\frac{7}{2} ) + 2 \ln (4) + 2 \ln (\frac{9}{2}) + 2 \ln (5) + 2 \ln (\frac{11}{2}) + 2 \ln ( 6) + 2 \ln (\frac{13}{2} ) + \ln 7] s u m = 7.603676$

Error

$f (x) = \ln (x) f' (x) = \frac{1}{x} f'' (x) = - \frac{1}{x^{2}} f'' (x) i s n e g a t i v e s o i t i s c o n c a v e d o w n w a r d o n t h e i n t e r v a l [1, 3] M = f'' (x) | = | - \frac{1}{x^{2}} | \leq | - \frac{1}{1^{2}} | \leq 1 | E | \leq \frac{M {(b - a)}^{3}}{12 n^{2}} | E | \leq \frac{1 {(7 - 1)}^{3}}{12 {(12)}^{2}} \leq 0.125 A c t u a l a r e a i s 7.6213 i s w i t h i n t h e e r r o r b o u n d o f a p p r o x i m a t e v a l u e$

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Short Answer

Step by step solution

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Definite integral

Trapezoidal sum

Error

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Calculate each definite integral approximation in Exercises 23–40, and then find an error bound for your approximation. If it is possible to calculate the definite integral exactly, then do so and verify that the error bounds you found are accurate∫17ln(x)dx,trapezoidsum,n=12

Short Answer

Step by step solution

Given Information

Definite integral

Trapezoidal sum

Error

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Decision Maths

Theoretical and Mathematical Physics

Pure Maths

Statistics

Logic and Functions

Study anywhere. Anytime. Across all devices.