Period 6, AP Calculus
Journals Section 7.1 and 7.2 Section 7.1 Section 7.1 was about how in many applications, the integral can be used to find net change over time. An important example of this is in distance traveled. By calculating the area of a velocity function using an integral, we can easily find the position, or distance of an object over a certain period of time. A key example of this is displacement. When velocity is constant during a motion, we can find the displacement (change in position) with the formula Displacement = Rate of change X time. This was seen back a couple of chapters where we used this simple formula to immediately yield us an answer. However, here, our velocity varies, so we resort instead to partitioning time intervals into subintervals so that the velocity is constant, and we can do so by using an integral. Also, in this chapter, we are given a specific initial value such as f(0) = 3. Thus, we must add our end result by 3 because it is our initial value and becomes added to the final answer. If the end result turns out negative, the particle will move left. On the other hand, if the end result turns out positive, the particle will move to the right. Another key concept in this chapter was finding total distance. By now, we know that to find the total distance, all we need to do is find the integral and solve for the area. However, in this chapter, it is not as easy. If we simply use the integral and solve for area, it is possible to get negative area, or area can be subtracted from the total which is not what we want. Instead, we must make sure we integrate within subintervals and add using the absolute value, so that area (a.k.a. Distance) is not subtracted so that we can achieve total distance rather than position. We must make sure we find where the graph shifts from positive to negative or negative to positive in order to do so.
1) The function v(t) is the velocity in m/sec of a particle moving along the x-axis. a) Determine when the particle is moving to the right, to the left, and stopped. b) Find the particle’s displacement for the given time interval. If s(0) = 3, what is the final position? c) Find the total distance traveled by the particle. 1. v(t) = 49 – 9.8t, 0 < t < 10
2. v(t) = 5cos t, 0 < t < 2pi
Section 7.2 We know how to find the area of a region between a curve and the x-axis but sometimes we want to know the area of the region bounded by one curve, y = f(x) and below by another, y = g(x). To do so, we need to approach this problem differently than finding the area below one curve and the x-axis. We need to integrate the curves in subintervals by partitioning them into several strips that we can individually calculate the area for using integrals. To do so, we may need more than two integrals all using different regions, and we may need to either subtract or add these integrals. To find the regions we need to use, we set the equations equal to each other and using algebra, we solve for…