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

472 Chapter 15 Applications of integration
f (t)
h
a b t
Figure15.1
Thearea underthe curve fromt =atot =bandthe
area ofthe rectangle are equal.
Example15.1 Findthe average value of f (t) =t 2 across
(a) [1,3]
(b) [2,5]
Solution (a) average value =
(b) average value =
∫ 3
1 t2 dt
3 −1 = 1 [ ] t
3 3
2 3
1
∫ 5
2 t2 dt
5 −2 = 1 [ ] t
3 5
3 3
2
= 13
3
= 13
Example15.1showsthatiftheintervalofintegrationchangesthentheaveragevalueof
a functioncan change.
Engineeringapplication15.1
Saw-toothwaveform
Recall from Engineering application 2.2 that engineers frequently make use of the
saw-toothwaveform. Consider the saw-toothwaveform shown inFigure 15.2.
y
5
–2 0 2
4 6 8 t
Figure15.2
Asaw-tooth waveform.
Calculate the average value of thiswaveform over a complete period.