5.1 Estimating with Finite Sums. Distance Traveled The distance traveled and the area are both found by multiplying the rate by the change in time.
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Would the area of this irregular region still give the total distance traveled over the time interval?
Newton and Leibniz (and other mathematicians) considered this question. They thought that it would and that is why they were interested in a calculus for finding areas under curves.
They argued that, just as the total area could be formed by summing the areas of the (essentially rectangular) strips, the total distance traveled could be found by summing the small distances traveled over the tiny time intervals.
We derive the area (1/4)(mi)² for each of the twelve subintervals and add them:
Since this number approximates the area and the total distance traveled by the particle, we conclude that the particle has moved approximately 9 units in 3 seconds. If it starts at x = 0, then it is very close to x = 0 when t = 3.
In Example 1, we used the Midpoint Rectangular Approximation Method (MRAM) to approximate the area under the curve.
The name suggests the choice we made when determining the heights of the approximating rectangles:
We evaluated the function at the midpoint of each subinterval.
If instead we had evaluated the function at the left-hand endpoints we would have obtained the LRAM approximation, and if we had used the right-hand endpoints we would have obtained the RRAM approximation.
Figure 5.8 shows the graph of f(x) = x2 sin x on the interval [0 , 3]. Estimate the area under the curve from x = 0 to x = 3.
It is not necessary to compute all three sums each time just to approximate the area, but we wanted to show again how all three sums approach the same number. With 1000 subintervals, all three agree in the first three digits. (The exact area is -7 cos 3 + 6 sin 3 – 2 which is 5.77666752456 to twelve digits.