EDM 107 - CALCULUS 2
CALCULATING ARC LENGTH
WITH THE USE OF INTEGRATION
LEARNING
OBJECTIVES
At the end of the lesson, the pre-service
math teachers will be able to:
• Recall the basic rule of differentiation (power
rule and exponential functions)
• Define arc length
• Solve the arc lengths using differentiation and
integration
• And solve real-life problems in solving the arc
length
REVIEW
POWER RULE OF DIFFERENTIATION
DERIVATIVE OF EXPONENTIAL FUNCTION
EDM 107 - CALCULUS 2
DEFINITION OF
ARC LENGTH
Let the function y = f (x) represents a smooth
curve on the interval (a, b). The arc length of f
between a and b is
EDM 107 - CALCULUS 2
Similarly, for a smooth curve x = g (y), the arc
length between c and d is
d
c
𝑑𝑥
𝑑𝑦
𝑑𝑦
EDM 107 - CALCULUS 2
Example:
Find the arc length of the graph of 𝑦 =
STEP 1: Find the derivative of f’(x)
𝑥7
14
+
1
10𝑥 5
on the interval 1,2
EDM 107 - CALCULUS 2
Example:
Find the arc length of the graph of 𝑦 =
STEP 2: Put in f’(x) in the formula.
𝑥3
6
+
1
2𝑥
on the interval
1
,2
2
EDM 107 - CALCULUS 2
Example:
Find the arc length of the graph of 𝑦 =
STEP 3: Simplify.
𝑥3
6
+
1
2𝑥
on the interval
1
,2
2
EDM 107 - CALCULUS 2
Example:
Find the arc length of the graph of 𝑦 =
STEP 4: Calculate the Integral.
𝑥3
6
+
1
2𝑥
on the interval
1
,2
2
EDM 107 - CALCULUS 2
Example:
Find the arc length of the graph of 𝑦 =
STEP 5: Calculate the integral by substituting the values of
the interval (upper and lower limit) and subtract. (Note:
make sure that the upper limit is subtracted to the lower
limit to get a positive answer)
.
𝑥3
6
+
1
2𝑥
on the interval
1
,2
2
EDM 107 - CALCULUS 2
Application Example :
An electric cable is hung between two towers that are 200 feet apart,
Figure 7.42. The cable takes the shape of a catenary whose equation is
𝑦 = 75 𝑒
𝑥
150
STEP 1: Transform the hyperbolic function into an
exponential function.
𝑒
𝑥
150
= 150
𝑥
cosh
150
EDM 107 - CALCULUS 2
Application Example :
An electric cable is hung between two towers that are 200 feet apart, Figure 7.42. The cable takes the shape
of a catenary whose equation is
𝑥
𝑥
𝑦 = 75 𝑒
150
𝑒
150
=
STEP 2: Find the derivative of the exponential function.
𝑥
150 cosh
150
EDM 107 - CALCULUS 2
Application Example :
An electric cable is hung between two towers that are 200 feet apart, Figure 7.42. The cable takes the shape
of a catenary whose equation is
𝑥
𝑥
𝑦 = 75 𝑒
150
STEP 3: Plug in the f’(x) in the arc length formula.
𝑒
150
=
𝑥
150 cosh
150
EDM 107 - CALCULUS 2
Application Example :
An electric cable is hung between two towers that are 200 feet apart, Figure 7.42. The cable takes the shape
of a catenary whose equation is
𝑥
𝑥
𝑦 = 75 𝑒
STEP 4: Simplify.
150
𝑒
150
=
𝑥
150 cosh
150
EDM 107 - CALCULUS 2
Application Example :
An electric cable is hung between two towers that are 200 feet apart, Figure 7.42. The cable takes the shape
of a catenary whose equation is
𝑥
𝑥
𝑦 = 75 𝑒
STEP 5: Take the root of the function.
150
𝑒
150
=
𝑥
150 cosh
150
EDM 107 - CALCULUS 2
Application Example :
An electric cable is hung between two towers that are 200 feet apart, Figure 7.42. The cable takes the shape
of a catenary whose equation is
𝑥
𝑥
𝑦 = 75 𝑒
STEP 6: Integrate.
150
𝑒
150
=
𝑥
150 cosh
150
EDM 107 - CALCULUS 2
Application Example :
An electric cable is hung between two towers that are 200 feet apart, Figure 7.42. The cable takes the shape
of a catenary whose equation is
𝑥
𝑥
𝑦 = 75 𝑒
STEP 7: Calculate the
integral by substituting
the values of the
interval (upper and
lower
limit)
and
subtract. (Note: make
sure that the upper
limit is subtracted to
the lower limit to get a
positive answer)
150
𝑒
150
=
𝑥
150 cosh
150
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