Chapter 5 | Integration
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Figure 5.33 The graph shows velocity versus time for a particle moving with a linear velocity function.
Example 5.25 Finding the Total Distance Traveled
Use Example 5.24 to find the total distance traveled by a particle according to the velocity function v ( t ) =3 t −5 m/sec over a time interval [0, 3].
Solution The total distance traveled includes both the positive and the negative values. Therefore, we must integrate the absolute value of the velocity function to find the total distance traveled. To continue with the example, use two integrals to find the total distance. First, find the t -intercept of the function, since that is where the division of the interval occurs. Set the equation equal to zero and solve for t . Thus, 3 t −5 = 0 3 t = 5 t = 5 3 . The two subintervals are ⎡ ⎣ 0, 5 3 ⎤ ⎦ and ⎡ ⎣ 5 3 , 3 ⎤ ⎦ . To find the total distance traveled, integrate the absolute value of the function. Since the function is negative over the interval ⎡ ⎣ 0, 5 3 ⎤ ⎦ , we have | v ( t ) | =− v ( t ) over that interval. Over ⎡ ⎣ 5 3 , 3 ⎤ ⎦ , the function is positive, so | v ( t ) | = v ( t ). Thus, we have
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