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Graphical Understanding of Integral |
CALCULUS
Higher Concepts of Mathematics
The value of this integral can be determined for
Figure 8 Graph of Velocity vs. Time
the case plotted in Figure 8 by noting that the
velocity is increasing linearly. Thus, the average
velocity for the time interval between tA and tB is
the arithmetic average of the velocity at tA and
the velocity at tB. At time tA, v = 6tA; at time tB,
v = 6tB. Thus, the average velocity for the time
interval between tA and tB is
which
6tA 6tB
2
equals 3(tA + tB). Using this average velocity, the
total distance traveled in the time interval
between tA and tB is the product of the elapsed
time tB - tA and the average velocity
3(tA + tB).
s = vavDt
s = 3(tA + tB)(tB - tA)
(5-16)
Equation 5-16 is also the value of the integral of the velocity, v, with respect to time, t, between
the limits tA -tB for the case plotted in Figure 8.
tB
tA
vdt 3(tA tB)(tB tA)
The cross-hatched area in Figure 8 is the area under the velocity curve between t = tA and t =
tB. The value of this area can be computed by adding the area of the rectangle whose sides are
tB - tA and the velocity at tA, which equals 6tA - tB, and the area of the triangle whose base is tB -
tA and whose height is the difference between the velocity at tB and the velocity at tA, which
equals 6tB - tA.
Area [(tB tA)(6tA)]
1
2
(tB tA)(6tb 6tA)
Area 6tA tB 6t2A 3t2B 6tA tB 3t2A
Area 3t2B 3t2A
Area 3(tB tA)(tB tA)
MA-05
Page 44
Rev. 0
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