Answered: y=1+cosx

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Related questions

Question

Approximating the shaded area with the

a) Trapezoidal rule and a step size of h=2 yields? .....in square units

b) Trapezoidal rule and a step size of h=1 yields? .....in square units

c) Romberg integration technique, with initial step size of h=2, yields? .....in square units

Expert Solution

Step 1

(a) $y = 1 + \cos x$

$h = 2$

and interval $[- 2, 2]$

according to trapezoidal rule

$\int_{a}^{b} f (x) d x \approx \frac{h}{2} [f (x_{0}) + 2 (f (x_{1}) + f (x_{2}) + . . . . f (x_{n - 1})) + f (x_{n})]$

$\int_{- 2}^{2} (1 + \cos x) d x \approx \frac{h}{2} [f (x_{0}) + 2 (f (x_{1})) + f (x_{2})] = \frac{2}{2} [(1 + \cos (- 2)) + 2 (1 + \cos (0)) + (1 + \cos (2))] = 1 [0.5839 + 2 \times 2 + 0.5839] = 5.1678$

area of the shaded region is $5.1678 sq units$

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