082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

13.3 Definite and indefinite integrals 447
y
0
p
3p ––2 y = sin x
x
Figure13.13
Thepositive andnegative areas are calculated
separately.
The total area is 3 square units. Note, however, that the single integral over 0 to 3π/2
evaluates to1;thatis, itgives the net value of 2 and −1.
∫ 3π/2
0
( ) 3π
sinx dx = [−cosx] 3π/2
0
=−cos +cos0=1
2
Theneedtoevaluatetheareaunderacurveisacommonrequirementinengineering.
Oftentherateofchangeofanengineeringvariablewithtimeisknownanditisrequired
to calculate the value of the engineering variable. This corresponds to calculating the
area under a curve.
Engineeringapplication13.3
Energyusedbyanelectricmotor
Consider a small d.c. electric motor being used to drive an electric screwdriver. The
amount of power that is supplied to the motor by the battery depends on the load on
the screwdriver. Therefore the power supplied to the screwdriver is a function that
varies with time.Figure 13.14shows a typicalcurve ofpower versus time.Now,
P = dE
dt
whereP = power (W),E = energy (J). Therefore, to calculate the energy used by
the motor between timest 1
andt 2
, wecan write
E =
∫ t2
t 1
Pdt
This is equivalent to evaluating the area under the curve, P(t), between t 1
and t 2
,
which is shown as the shaded region inFigure 13.14.
Power (watts)
t 1
t 2
Time (s)
Figure13.14
Shaded area representsthe energy usedto
drivethe motor duringthe time interval
t 1 tt 2 .