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Length of an Arc - Engineering Mathematics, Lecture notes of Engineering Mathematics

FEU Institute of Technology (FEUIT)Engineering Mathematics

All the topics that the Length of an Arc will cover. Unlock it and see it yourself; I will ensure it will help you.

Typology: Lecture notes

2021/2022

Available from 08/29/2022

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LENGTH OF AN ARC
Let the function given by y=f(x) represent a smooth curve on the interval .
The arc length of the function f between a and b is,
𝑆=
∫
𝑎
𝑏
√
1+
(
𝑦′
)
2
𝑑𝑥
Similarly, for the smooth curve given by x=g(y), arc length of function g between c and d is,
𝑆=
∫
𝑐
𝑑
√
1+
(
𝑥′
)
2
𝑑𝑦
pf3
pf4

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LENGTH OF AN ARC

Let the function given by y=f(x) represent a smooth curve on the interval.

The arc length of the function f between a and b is,

𝑎 𝑏

√^1 +(^ 𝑦^ ′^ )

2

Similarly, for the smooth curve given by x=g(y), arc length of function g between c and d is,

𝑐 𝑑

√^1 +(^ 𝑥^ ′^ )

2

Determine the length of y=ln(secx) between

In this case we’ll need to use the first ds since the function is in the form y=f(x). So, let’s get the derivative out of

the way.

𝑦 =ln 𝑠𝑒𝑐𝑥

′

2

2 𝑆=∫ 0 𝜋 4 √^1 +(^ 𝑦^ ′^ ) 2 𝑑𝑥 𝑆=∫ 0 𝜋 4 √^1 +(^ 𝑡𝑎𝑛𝑥^ ) 2 𝑑𝑥 𝑆=∫ 0 𝜋 4 √ 𝑠𝑒𝑐 2 𝑥 𝑑𝑥 𝑆=∫ 0 𝜋 4 sec 𝑥 𝑑𝑥 𝑆=𝑙𝑛|𝑠𝑒𝑐𝑥+𝑡𝑎𝑛𝑥| 0 𝜋 4 𝑆=𝑙𝑛√ 2 + 1 𝑢𝑛𝑖𝑡𝑠