文档介绍：Section
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Distance, Position, and
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Although we motivated the definition of the definite integral with the notion of area, there
are many applications of integration to problems unrelated to putation of area.
Depending on the context, the definite integral of a function f from a to b could represent
the total mass of a wire, the total electric charge on such a wire, or the probability that a
light bulb will fail sometime in the time interval from a to b. In this section we will consider
three applications of definite integrals: finding the distance traveled by an object over an
interval of time if we are given its velocity as a function of time, finding the position of an
object at any time if we are given its initial position and its velocity as a function of time,
and finding the length of a curve.
Distance
Suppose the function v is continuous on the interval [a, b] and, for any a ≤ t ≤ b, v(t)
represents the velocity at time t of an object traveling along a line. Divide [a, b] into n
time intervals of equal length
b − a
∆t =
n
with endpoints a = t0 < t1 < t2 < · · · < tn = b. Then, for j = 1, 2, 3, . . . , n, |v(tj−1)| is the
speed of the object at the beginning of the jth time interval. Hence, for small enough ∆t,
|v(tj−1)|∆t will give a good app