Explanation Given: The velocity v ( t ) is given by v ( t ) = 1 + 2 t . The time interval is given by [ 1 , 2 ] Number of sub-intervals is n = 4 Formula used: Formula to approximate the value of total distance traveled by an object moving with a velocity v ( t ) using Riemann sum for right end points and n sub-intervals is given by S n = ∑ k = 1 n | v ( a + k Δ t ) | ( Δ t ) Formula to approximate the value of total distance traveled by an object moving with a velocity v ( t ) using Riemann sum for right end points and n sub-intervals is given by S n = ∑ k = 1 n | v [ a + ( k − 1 ) Δ t ] | ( Δ t ) Where Δ t = b − a n Calculation: Consider the given velocity function. v ( t ) = 1 + 2 t Here, the interval is [ 1 , 2 ] . Therefore, b = 2 , a = 1 and n = 4 . Therefore, width of each sub-interval will be as shown. Δ t = b − a n = 2 − 1 4 = 1 4 = 0.25 Now, the points will be as follows: t 0 = a + 0 ( Δ t ) = 1 + 0 = 1 t 1 = 1 + Δ t = 1 + 0.25 = 1.25 t 2 = t 1 + Δ t = 1.25 + 0.25 = 1.50 t 3 = 1.75 t 4 = 2.0 Now, approximate the value of S n = ∑ k = 1 n | v ( a + k Δ t ) | ( Δ t ) for n = 4 . S 4 = ∑ k = 1 4 | v ( a + k Δ t ) | ( Δ t ) = ( Δ t ) | [ v ( t 1 ) + v ( t 2 ) + v ( t 3 ) + v ( t 4 ) ] | = ( 0

7th Edition

ISBN: 9781524916817

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Differential Equations

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Chapter 5.3, Problem 28PS

To determine

To calculate:The estimated value of total distance traveledby the object during given time interval using Riemann sum for 4 subintervals. Also compare the result for left end points and right end points as well.

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Chapter 5.3, Problem 27PS

Chapter 5.3, Problem 29PS

Chapter 5 Solutions

Calculus

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The estimated value of total distance traveledby the object during given time interval using Riemann sum for 4 subintervals. Also compare the result for left end points and right end points as well.

Solution Summary: The author calculates the estimated value of total distance traveled by the object during given time interval using Riemann sum for 4 subintervals.

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To calculate:The estimated value of total distance traveledby the object during given time interval using Riemann sum for 4 subintervals. Also compare the result for left end points and right end points as well.

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Chapter 5 Solutions