Explanation Given: The integral is ∫ 0 π 4 ( 3 sin 2 x ) d x = 3 2 . Definition used: 1 Suppose f is defined and integrable on [ a , b ] . The Midpoint Rule approximation to ∫ a b f ( x ) d x using n equally spaced subintervals on [ a , b ] is, M ( n ) = ∑ k = 1 n f ( x k − 1 + x k 2 ) Δ x . Where, Δ x = b − a n and x 0 = a , x k = a + k Δ x , for k = 1 , 2 , 3... n . Definition used: 2 Suppose f is defined and integrable on [ a , b ] . The Trapezoid Rule approximation to ∫ a b f ( x ) d x using n equally spaced subintervals on [ a , b ] is, T ( n ) = ( 1 2 f ( x 0 ) + ∑ k = 1 n − 1 f ( x k ) + 1 2 f ( x n ) ) Δ x . Where, Δ x = b − a n and x 0 = a , x k = a + k Δ x , for k = 1 , 2 , 3... n . Calculation: The given integral is ∫ 0 π 4 ( 3 sin 2 x ) d x = 3 2 which is equal to 1.5. Consider the case n = 4 . Compute the value of Δ x . Δ x = b − a n = 0.7854 − 0 4 = 0.7854 4 = 0.19635 Compute the value of x k . x k = a + k Δ x = 0 + k ( 0.19635 ) = k ( 0.19635 ) , for k = 0 , 1 , 2 , 3 , 4 Use midpoint formula and online calculator to compute the value of M ( 4 ) . M ( 4 ) = ∑ k = 1 4 f ( ( k ( 0.19635 ) − 1 ) + ( k ( 0.19635 ) ) 2 ) Δ x = ∑ k = 1 4 f ( 2 k ( 0.19635 ) − 1 2 ) ( 0.19635 ) ≈ 1.509682 [ Use online calculator ] Use Trapezoid formula and online calculator to compute the value of T ( 4 ) . T ( 4 ) = ( 1 2 f ( 0 ) + ∑ k = 1 3 f ( − 2 ( 1 − k ( 0.19635 ) ) ) + 1 2 f ( 1.5 ) ) ( 0.19635 ) ≈ 1.4806737 Use absolute error formula to compute the error in the Midpoint Rule approximation for n = 4 . | M ( 4 ) − 1.5 | = | 1.509682 − 1.5 | ≈ 9.6 × 10 − 3 Use absolute error formula to compute the error in the Trapezoid Rule approximation for n = 4 . | T ( 4 ) − 1.5 | = | 1.4806737 − 1.5 | ≈ 1.9 × 10 − 2 Consider the case n = 8 . Compute the value of Δ x . Δ x = b − a n = 0.7854 − 0 8 = 1.5 8 = 0.098175 Compute the value of x k . x k = a + k Δ x = 0 + k ( 0.098175 ) = k ( 0.098175 ) , for k = 0 , 1 , 2 , 3 , 4 Use midpoint formula and online calculator to compute the value of M ( 8 ) . M ( 8 ) = ∑ k = 1 8 f ( 1 + ( ( k ( 0.098175 ) − 1 ) ) + 1 + k ( 0.098175 ) 2 ) Δ x = ∑ k = 1 8 f ( ( k ( 0.098175 ) − 1 + k ( 0.098175 ) ) ( 1 ) + 2 2 ) ( 0.098175 ) = ∑ k = 1 8 f ( ( 2 k ( 0.098175 ) − 1 ) + 2 ) 2 ( 0.098175 ) ≈ 1.502413 [ Use online calculator ] Use Trapezoid formula and online calculator to compute the value of T ( 8 ) . T ( 8 ) = ( 1 2 f ( 0 ) + ∑ k = 1 7 f ( 1 + k ( 0.098175 ) ) + 1 2 f ( 0

Calculus, Single Variable: Early Transcendentals (3rd Edition)

3rd Edition

ISBN: 9780134766850

Author: William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz

Publisher: PEARSON

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Chapter 8.8, Problem 31E

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To make: A table to show the approximation and errors for n=4,8,16,32.

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Chapter 8.8, Problem 30E

Chapter 8.8, Problem 32E

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Instructions: Calculate the definite integral of: (Image) I will be grateful for your help :)

What trig substitution is correct for : -dx 7 tan e 7 sine 49sin e 7 sece 49 tan e 49sece V49- . Write the integral in terms of e for f -dx using the answer you 4x chose in #5. DO not simplify. What is the value of the integral 49- dr in terms of 0? Use your answer from #6 to begin your work. What is the value of the integral (V49- x² dx in terms of x? Use your 4x result from #7 to answer this question. Use a right triangle or other means to show your work.

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Chapter 8 Solutions

Calculus, Single Variable: Early Transcendentals (3rd Edition)

Ch. 8.1 - What change of variable would you use for the...Ch. 8.1 - Prob. 2QC Ch. 8.1 - Prob. 3QC Ch. 8.1 - Prob. 4QC Ch. 8.1 - What change of variables would you use for the...Ch. 8.1 - Prob. 2E Ch. 8.1 - What trigonometric identity is useful in...Ch. 8.1 - Prob. 4E Ch. 8.1 - Prob. 5E Ch. 8.1 - Prob. 6E

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To make: A table to show the approximation and errors for n = 4 , 8 , 16 , 32 .

Solution Summary: The author describes the Midpoint Rule approximation and errors for n=4,8,16,32.

To make: A table to show the approximation and errors for n=4,8,16,32.

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Chapter 8 Solutions