Explanation Given information: The given integral is ∫ 0 1 e − x 2 d x . The value of N is 5. Formula Used: The integral is in the form of ∫ a b f ( x ) d x . The N th trapezoidal approximation is: T N = 1 2 Δ x ( x 0 + 2 x 1 + … + 2 x ( N − 1 ) + x N ) Where, Δ x = b − a N and x j = f ( a + j Δ x ) . The formula for the midpoint rule M N is: M N = Δ x ( f ( c 1 ) + f ( c 2 ) + … + f ( c N ) ) Where, Δ x = b − a N and c j = a + ( j − 1 2 ) Δ x . Calculation: The integral is I = ∫ 0 1 e − x 2 d x . Compare the given integral with ∫ a b f ( x ) d x . So, the function be f ( x ) = e − x 2 . And, a = 0 , b = 1 . Find the value of Δ x . Δ x = b − a N = 1 − 0 5 = 1 5 To find the value of T 5 , first find the values of x 0 , x 1 , x 2 , x 3 , x 4 and x 5 . Substitute 0 for j , 1 5 for Δ x and 0 for a in the formula x j = f ( a + j Δ x ) . x 0 = f ( 0 + 0 ( 1 5 ) ) = f ( 0 ) = e − ( 0 ) 2 = 1 Substitute 1 for j , 1 5 for Δ x and 0 for a in the formula x j = f ( a + j Δ x ) . x 1 = f ( 0 + 1 ( 1 5 ) ) = f ( 0 + 1 5 ) = e − ( 1 5 ) 2 = 0.9608 Substitute 2 for j , 1 5 for Δ x and 0 for a in the formula x j = f ( a + j Δ x ) . x 2 = f ( 0 + 2 ( 1 5 ) ) = f ( 0 + 2 5 ) = e − ( 2 5 ) 2 = 0.8521 Substitute 3 for j , 1 5 for Δ x and 0 for a in the formula x j = f ( a + j Δ x ) . x 3 = f ( 0 + 3 ( 1 5 ) ) = f ( 0 + 3 5 ) = e − ( 3 5 ) 2 = 0.6977 Substitute 4 for j , 1 5 for Δ x and 0 for a in the formula x j = f ( a + j Δ x ) . x 4 = f ( 1 + 4 ( 1 5 ) ) = f ( 1 + 4 5 ) = e − ( 4 5 ) 2 = 0.5273 Substitute 5 for j , 1 5 for Δ x and 0 for a in the formula x j = f ( a + j Δ x ) . x 5 = f ( 0 + 5 ( 1 5 ) ) = f ( 0 + 1 ) = e − ( 1 ) 2 = 0.3679 Now, substitute 1 5 for Δ x , 1 for x 0 , 0.9608 for x 1 , 0.8521 for x 2 , 0.6977 for x 3 , 0.5273 for x 4 and 0.3679 for x 5 in the expression T N = 1 2 Δ x ( x 0 + 2 x 1 + … + 2 x ( N − 1 ) + x N ) . T 5 = 1 2 ⋅ 1 5 ( 1 + 2 ( 0.9608 ) + 2 ( 0.8521 ) + 2 ( 0.6977 ) + 2 ( 0.5273 ) + 0.3679 ) = 1 10 ( 1 + 1.9216 + 1.7042 + 1.3954 + 1.0546 + 0

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

See similar textbooks

Videos

Question

Chapter 8.9, Problem 11E

To determine

To calculate:

The value of MN and TN

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 8.9, Problem 10E

Chapter 8.9, Problem 12E

Students have asked these similar questions

5 2. Compute the value of V25–x² dx

9. 00 J2 e -5p dp

Pn 1995 Hhe moose papulalion n MeaSured to be 1900.TTBa, 1994, Hbe Was, eaSuned agaio to be Uu400.1f Hbe Cantinves te papulation in to Pagulatien charng farmula for the maoseppsalian „P,in Ferms of t, the 6, Snce 1990.

Chapter 8 Solutions

Calculus - Standalone book

Ch. 8.1 - Prob. 1PQ Ch. 8.1 - Prob. 2PQ Ch. 8.1 - Prob. 3PQ Ch. 8.1 - Prob. 1E Ch. 8.1 - Prob. 2E Ch. 8.1 - Prob. 3E Ch. 8.1 - Prob. 4E Ch. 8.1 - Prob. 5E Ch. 8.1 - Prob. 6E Ch. 8.1 - Prob. 7E

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.