Q 34 Calculate each definite integral... [FREE SOLUTION]

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Chapter 5: Q 34 (page 492)

Calculate each definite integral approximation in Exercises 23鈥�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
${鈭�}_{0}^{3} \cos (x^{2}) d x, m i d p o i n t s u m, n = 9$

Short Answer

Expert verified

$S u m_{m i d p o i n t} = 0.716498 E r r o r = 0.15277$

Step by step solution

Given Information

${鈭�}_{0}^{3} \cos (x^{2}) d x, m i d p o i n t s u m, n = 9$

Definite Integral

${鈭�}_{0}^{3} \cos (x^{2}) d x D e f i n i t e i n t e g r a l v a l u e w i l l b e c a l c u l a t e d u \sin g c a l c u l a t o r {鈭�}_{0}^{3} \cos (x^{2}) d x = 0.702864$

Midpoint sum

${鈭�}_{a}^{b} f (x) d x = 鈭� x [f (\frac{x_{0} + x_{1}}{2}) + f (\frac{x_{1} + x_{2}}{2}) + . . . . . + f (\frac{x_{n - 1} + x_{n}}{2})] 鈭� x = \frac{b - a}{n} = \frac{3 - 0}{9} = \frac{1}{3} M i d p o i n t s a r e (0 + \frac{1}{3}), (\frac{1}{3} + \frac{2}{3}) 鈥�, (\frac{2}{3} + 1), (1 + \frac{4}{3}), (\frac{4}{3} + \frac{5}{3}) 鈥�, (\frac{5}{3} + 2) 鈥�, (2 + \frac{7}{3}), (\frac{7}{3} + \frac{8}{3}) 鈥�, (\frac{8}{3} + 3) 鈥� (\frac{1}{6}), (\frac{1}{2}), (\frac{5}{6}), (\frac{7}{6}), (\frac{3}{2}), (\frac{11}{6}) . (\frac{13}{6}), (\frac{5}{2}), (\frac{17}{6}) S u m = \frac{1}{3} [\cos {(\frac{1}{6})}^{2} + \cos {(\frac{1}{2})}^{2} + \cos {(\frac{5}{6})}^{2} + \cos {(\frac{7}{6})}^{2} + \cos {(\frac{3}{2})}^{2} + \cos {(\frac{11}{6})}^{2} + \cos {(\frac{13}{6})}^{2} + \cos {(\frac{5}{2})}^{2} + \cos {(\frac{17}{6})}^{2}] S u m = 0.716498$

Error

Error bound of midpoint sum is

$f (x) = \cos (x^{2}) f'' (x) = - 4 (x) \cos (x^{2}) f'' (3) = 11 | E r r o r | 鈮� \frac{M {(b - a)}^{3}}{24 n^{2}} | E r r o r | 鈮� \frac{11 {(3 - 0)}^{3}}{24 {(9)}^{2}} E r r o r 鈮� 0.1527$

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Short Answer

Step by step solution

Given Information

Definite Integral

Midpoint sum

Error

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Calculate each definite integral approximation in Exercises 23鈥�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 鈭�03cos(x2)dx,midpointsum,n=9

Short Answer

Step by step solution

Given Information

Definite Integral

Midpoint sum

Error

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Geometry

Applied Mathematics

Logic and Functions

Decision Maths

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.