Chabot Mathematics
§8.3 Trig
Integral Apps
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx
Review §
8.2
 Any QUESTIONS About
• §8. → Trigonometric
Derivatives
 Any
QUESTIONS
About HomeWork
• §8.2 → HW-11
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx
§8.3 Learning Goals
 Derive and use integration formulas
for trigonometric functions
 Apply integrals of periodic functions
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx
Trigonometric AntiDerivatives
 Recall the Trig Derivs
d
sin u   cosu 
du
d
tan u   sec 2 u 
du
d
cosu    sin u 
du
d
secu   tan u   secu 
du
 Then the Trig AntiDerivatives
 cosu du  sin u   C
2
sec
 u du  tan u   C
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 sin u du   cosu   C
 tan u  secu du  secu   C
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx
Quick Example  Trig AnitDeriv
 Find
sec 2  1t 
Rt   
dt
2
AntiDerivative:
t
 SOLUTION:
• There is no formula available for the
immediate AntiDifferentiation of this
function, but we observe that the argument
of the secant function (i.e., the expression
1/t) has a derivative which is present in the
integrand.
– This makes SUBSTITUTION a likely choice
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx
Quick Example  Trig AnitDeriv
 For the Substitution, let: u  1t
 Next Isolate dt
1
u

t
du
1
 2
dt
t
d 
1
u  

dt 
t
1  du
 du
   2
t  1
 dt
d
d 1
u
dt
dt t
1
 du   2 dt
t
1  2

2


du


dt

t


t
du  dt
2


t

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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx
Quick Example  Trig AnitDeriv
 Substitute for t & dt then Take
AntiDerivative
sec 2  1t 
sec 2 u 
2
dt



t
du
 t2
 t2


   sec u  du
2
  tan u   C
  tan  1t   C
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx
Example  Cyclical Sales
 A product is initially quite popular and
then settles into cyclical demand. The
demand now changes at an
instantaneous rate of
3
Rt  
 sin 0.12t   1
t 1
• Where
– R is the Sales Rate in kUnits per week
– t is time in the number of weeks after Product
Introduction
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx
Example  Cyclical Sales
 Use the Model to determine How many
units are sold in the second month after
release (assuming 4.5-week months)
 SOLUTION:
 To find an expression for the total sales
during the second month, find the value
of the definite integral over Month-2
 3

S t  2   Rt  dt   
 sin 0.12t   1 dt
t 1

4.5
4.5
9
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx
Example  Cyclical Sales
 Integrate Term-by-Term
 3

S 2   
 sin 0.12t   1 dt 
t 1

4.5
9
9
9
3


4.5 t  1 dt  4.5sin 0.12t  dt  4.51 dt
9
 Use TWO Separate Substitutions
du
u  t 1 
 1  du  dt
dt
dv
dv
v  0.12t 
 0.12 
 dt
dt
0.12
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx
Example  Cyclical Sale
 Then S 2
t 9
3
    du 
u
t  4.5  
t 9
 sin v 

 0.12  dv 
t  4.5
t 94.5
 Performing the Integrations
S 2 
S 2 
3 ln u 
t 9
t  4.5
3 ln t  1 
t 9
t  4.5
t 9
 cos v 
 

 0.12  t 4.5

 cos0.12t 
 

0
.
12

 t 4.5
9  4.5
t 9
 4.5
1
cos0.12  9  cos0.12  4.5  4.5
 3ln 9  1  ln 4.5  1  
0.12
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx
Example  Cyclical Sale
 Doing the Calculations
1
0.4713  0.8577  4.5
S 2  32.302  1.705 
0.12
 So Finally S 2 
9
 Rt  dt
 9.513
4.5
 Thus During the second month,
approximately 9,513 items are sold
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx
Check by MATLAB MuPAD
Integrand := 3/(t+1) + sin(12*t/100) + 1
S_of_t := int(Integrand, t)
Snum := numeric::int(3/(t+1) + sin(0.12*t) + 1, t=4.5..9)
Plot the AREA under the Integrand Curve
fArea := plot::Function2d(Integrand,
t = 4.5..9, GridVisible = TRUE):
plot(plot::Hatch(fArea), fArea,
Width = 320*unit::mm,
Height = 180*unit::mm,
AxesTitleFont = ["sans-serif", 24],
TicksLabelFont=["sans-serif", 16],
LineWidth = 0.04*unit
::inch,BackgroundColor =
RGB::colorName([0.8, 1, 1]) )
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx
Exponential·Trigonometric
 Integration formulas for the Products of
Exponentials and Sinusoids:
e
au
sin bu du 
au
e
 cosbu du 
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e au
a sin bu   b cosbu   C
2
2
a b
e au
a cos nbu   b sin bu   C
2
2
a b
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx
Example  Periodic-Fund F.V.
 A study suggests that investment in
equity funds varies in part according to
the effects of Seasonal Affect Disorder.
 Where
 A model for the
continuous rate of
• I(t) ≡ investment
rate in $M/year
Investment in a
particular market
• t ≡ time in years
after the Spring of
 
I t   4  cos  t 
Calendar Year
6 
2010
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx
Example  Periodic-Fund F.V.
 For this Fund Model find the future value
of the market’s investments after 10 years
for a prevailing interest rate of 4%
 SOLUTION:
 The future value of a continuous income
stream f(t) invested for T years at an
annual rate-of-return, r :
FV T   e rT
t T

f t e rt dt
t 0
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx
Example  Periodic-Fund F.V.
 For T = 10 and r = 0.04 (4%)
FV 10  e
t 10
0.04(10)

   0.04t
t 0 4  cos 6 t e dt
 0.04t 0.04t   
 e  4e
e
cos t  dt
 6 
0
10
0.4
 4
e
 0.04t
e 
e

2
 2

0
.
04

(

0
.
04
)

6

 0.04t
0.4
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10

  
   
 0.04 cos 6 t   6 sin  6 t  
 
   0

Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx
Example  Periodic-Fund F.V.
 Continuing the Calculation
0.04(10)

4
e
 e0.4 
e 0.04(10) 
2
 2

0
.
04

(

0
.
04
)

6

0.04( 0 )

4
e
 e0.4 
e 0.04( 0) 
2
 2

0
.
04

(

0
.
04
)

6



 

 
 0.04 cos 6 10   6 sin  6 10  



 


  
   
 0.04 cos 6  0   6 sin  6  0  



 

 Doing the Arithmetic find: FV  47.682
• Thus After 10 years of continuous
investment, the market will accrue about
$47,682,000 (compared to the ~$38.3M of
its own money that was invested).
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx
WhiteBoard Work
 Problems From §8.3
• P8.3-51 →
Heating
Degree
Days
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx
All Done for Today
Trig
Anti
Derivs
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
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 sin udu   cos u  c
 cos udu  sin u  c
 tan udu   ln cos u  c
 cot udu  ln sin u  c
 sec udu  ln sec u  tan u  c
 csc udu   ln csc u  cot u  c
 sec udu  tan u  c
 sec u tan udu  sec u  c
 cot u csc udu   csc u  c
 csc udu   cot u  c
2
2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx
Chabot Mathematics
Appendix
r  s  r  s r  s 
2
2
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
2a
–
Chabot College Mathematics
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BMayer@ChabotCollege.edu
2b
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx
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BMayer@ChabotCollege.edu • MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx
Plot Function
Hoft := 25 + 22*cos(2*PI*(t-35)/365)
plot(Hoft, t =0..365, GridVisible = TRUE,
LineWidth = 0.04*unit::inch,
Width = 320*unit::mm, Height = 180*unit::mm,
AxesTitleFont = ["sans-serif", 24],
TicksLabelFont=["sans-serif", 16])
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx
Verify Average Calculation
Hoft := 25 + 22*cos(2*PI*(t-35)/365)
Have := int(Hoft, t=0..90)/90
Havenum := float(Have)
Plot the H(t) Function over 0→365 days
plot(Hoft, t =0..365, GridVisible = TRUE,
LineWidth = 0.04*unit::inch,
Width = 320*unit::mm, Height = 180*unit::mm,
AxesTitleFont = ["sans-serif", 24],
TicksLabelFont=["sans-serif", 16])
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx