Chapter 5 | Integration
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velocity of an object as a function of time, then the area under the curve tells us how far the object is from its original position. This is a very important application of the definite integral, and we examine it in more detail later in the chapter. For now, we’re just going to look at some basics to get a feel for how this works by studying constant velocities. When velocity is a constant, the area under the curve is just velocity times time. This idea is already very familiar. If a car travels away from its starting position in a straight line at a speed of 75 mph for 2 hours, then it is 150 mi away from its original position ( Figure 5.20 ). Using integral notation, we have ∫ 0 2 75 dt =150.
Figure 5.20 The area under the curve v ( t ) =75 tells us how far the car is from its starting point at a given time.
In the context of displacement, net signed area allows us to take direction into account. If a car travels straight north at a speed of 60 mph for 2 hours, it is 120 mi north of its starting position. If the car then turns around and travels south at a speed of 40 mph for 3 hours, it will be back at it starting position ( Figure 5.21 ). Again, using integral notation, we have ∫ 0 2 60 dt + ∫ 2 5 −40 dt =120−120 =0. In this case the displacement is zero.
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