Explanation Given information: ∫ 0 π / 4 sec 2 x d x , n = 10 . Explanation: To calculate the exact value of the integral ∫ 0 π / 4 sec 2 x d x which is equivalent to ∫ 0 π / 4 ( 1 cos ( x ) ) 2 d x Step 1: Press [MATH][9] to access fnInt. Step 2: Press number 0 as a lower limit and Press [ALPHA] [^] to get π then press [ ÷ ] and a number 4 as an upper limit. Step 3: Press [ ( ] 1 [ ÷ ][COS] [ ( ][ALPHA] [STO>] [ ) ][ )] [^] 2. It will look likes: ∫ 0 π / 4 ( 1 / cos ( x ) ) 2 d _ Step 4: In front of d enter ALPHA] [STO>]. It will look likes: ∫ 0 π / 4 ( 1 / cos ( x ) ) 2 d x Step 5: Press [ENTER] to get the final answer as 1 . Thus, the exact value of ∫ 0 π / 4 ( 1 / cos ( x ) ) 2 d x is 1 . MIDPOINT RULE: The format for the midpoint rule used on the calculator is sum ( seq ( f ( x ) ) , X , a + Δ x 2 , b − Δ x 2 , Δ x ) ⋅ Δ x For ∫ 0 π / 4 ( 1 cos ( x ) ) 2 d x , n = 10 , Δ x = b − a n = π 4 − 0 10 = π 40 Start point: a + Δ x 2 = 0 + π / 40 2 = π 80 End point: b − Δ x 2 = π 4 − π / 40 2 = π 4 − π 80 = 19 π 80 To approximate the value of the integral ∫ 0 π / 4 ( 1 cos ( x ) ) 2 d x using a calculator follow the below steps: Step 1: From the home screen press [2ND][STAT] to access LIST, then go to [MATH][5]. It will look like: sum( 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: ( 1 cos ( x ) ) 2 Variable: X Start: π 80 End: 19 π 80 Step: π 40 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 ( ( 1 cos ( x ) ) 2 , X , π 80 , 19 π 80 , π 40 ) ) Step 3: Press 2 ND and press (-) to get Ans, and then multiply it by π 40 . The final input will look like sum ( seq ( ( 1 cos ( x ) ) 2 , X , π 80 , 19 π 80 , π 40 ) ) ∗ π 40 Step 4: Press [ENTER] to get the final answer as ≈ 0.8533236715 . Thus, ∫ 0 π / 4 sec 2 x d x ≈ 0.8533236715 using the midpoint rule Finding the error which is given by the formula: | Approximated value - Actual value | ⇒ | 0.8533236715 − 1 | ⇒ | − 0.146676329 | . ⇒ 0.146676329 Error using the midpoint rule is: 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 39E

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The approximate value of the integral ∫0π/4sec2 x dx by using the midpoint rule, the trapezoidal rule, and the Simpson’s rule with n = 10, then find the exact value by integration and give error for each approximation.

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Chapter 9.4, Problem 38E

Chapter 9.4, Problem 40E

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The approximate value of the integral ∫ 0 π / 4 sec 2 x d x by using the midpoint rule, the trapezoidal rule, and the Simpson’s rule with n = 10 , then find the exact value by integration and give error for each approximation.

Solution Summary: The author calculates the approximate value of the integral displaystyle 'underset' by using the midpoint rule, the trapezoidal rule and the Simpson's rule.

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The approximate value of the integral ∫0π/4sec2 x dx by using the midpoint rule, the trapezoidal rule, and the Simpson’s rule with n = 10, then find the exact value by integration and give error for each approximation.

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Chapter 9 Solutions