Explanation Given information: ∫ 0 1 2 x e x 2 d x , n = 10 . Explanation: To find exact value of the integral ∫ 0 1 2 x e x 2 d x using TI-84 calculator use the steps given below. Step 1: Press [MATH] and press 9 to access fnInt. Step 2: Press number 0 as a lower limit and 1 as an upper limit. Step 3: Enter a number 2 then [ALPHA] [STO>] [2 nd ] [ ÷ ] [^][ALPHA] [STO>] [^] 2 . It will look likes: ∫ 0 1 2 x e x 2 d _ Step 4: In front of d enter ALPHA] [STO>]. It will look likes: ∫ 0 1 2 x e x 2 d x Step 5: Press [ENTER] to get the final answer as 1.718281828 . Thus, the exact value of ∫ 0 1 2 x e x 2 d x is 1.718281828 . Midpoint Rule: The format for midpoint rule used on TI-84 is sum ( seq ( f ( x ) ) , X , a + Δ x 2 , b − Δ x 2 , Δ x ) ⋅ Δ x For ∫ 0 1 2 x e x 2 d x , n = 10 , Δ x = b − a n = 1 − 0 10 = 1 10 = 0.1 Start point: a + Δ x 2 = 0 + 1 / 10 2 = 1 20 = 0.05 End point: b − Δ x 2 = 1 − 1 / 10 2 = 1 − 1 20 = 1 − 0.05 = 0.95 To approximate the value of the integral ∫ 0 1 2 x e x 2 d x using TI-84 follow the below steps: Step 1: From the home screen press [2 nd ][STAT] to access LIST then go to [MATH] and press. Step 2: Press [2 nd ][STAT] to access LIST then go to [OPS] and press 5 to get seq the page Fill the below parameters by pressing [ENTER]: EXpr: 2 x e x 2 Variable:X Start: 0.05 End: 0.95 Step: 0.1 Paste Step 3: Move the cursor on “Paste” and Press [ENTER] and complete the bracket using [ )]. The output will display on a screen like: sum ( seq ( 2 x e x 2 , X , 0.05 , 0.95 , 0.1 ) ) Step 3: Press 2 ND and press (-) to get Ans and then multiply it by 0.1 . The final input will look like sum ( seq ( 2 x e x 2 , X , 0.05 , 0.95 , 0.1 ) ) ∗ 0.1 Step 4: Press [ENTER] to get the final answer as ≈ 1.712342989 . Thus, ∫ 0 1 2 x e x 2 d x ≈ 1.712342989 by using the Midpoint rule. Find error using formula: | Approximated value - Actual value | ⇒ | 1.712342989 − 1.718281828 | ⇒ | − 0.005938839 | . ⇒ 0.005938839 Thus, error using Midpoint rule is 0.005938839 . Trapezoidal Rule: The format for trapezoidal rule used on TI-84 is ( Calculate first term saperately + sum ( seq ( 2 f ( x ) ) , X , a + Δ x , b − Δ x , Δ x ) + Calculate second term saperately ) ⋅ Δ x 2 For ∫ 0 1 2 x e x 2 d x , n = 10 , Δ x = b − a n = 1 − 0 10 = 0

Calculus & Its Applications (14th Edition)

14th Edition

ISBN: 9780134437774

Author: Larry J. Goldstein, David C. Lay, David I. Schneider, Nakhle H. Asmar

Publisher: PEARSON

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Chapter 9.4, Problem 40E

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The approximate value of the integral ∫012xex2dx by using the Midpoint rule, Trapezoidal rule and Simpson’s rule with n = 10 then find the exact value by integration .Find error for each approximation.

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Chapter 9.4, Problem 39E

Chapter 9.4, Problem 41E

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The approximate value of the integral ∫ 0 1 2 x e x 2 d x by using the Midpoint rule, Trapezoidal rule and Simpson’s rule with n = 10 then find the exact value by integration .Find error for each approximation.

Solution Summary: The author explains how to determine the approximate value of the integral displaystyle 'underset'0overset1int, using the TI-84 calculator.

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The approximate value of the integral ∫012xex2dx by using the Midpoint rule, Trapezoidal rule and Simpson’s rule with n = 10 then find the exact value by integration .Find error for each approximation.

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Chapter 9 Solutions