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2.6
2.6.1
Integrals of functions with radicals, and the length of curves.
Case Study: The Golden Gate Bridge
The Golden Gate Bridge is a remarkable engineering feat and one of the most beautiful bridges in the
world. It was designed in the ealy 30s and completed in 1937. The shape of the cables between the two
main bridge-posts is a perfect parabola; the parabola is the shape naturaly adopted by hanging cables when
they are “carrying” an equally distributed weight.
What is the equation of this parabola?
We now turn to the question “What is the total length of the arch (from one base to the other)?”. In order
to answer this question, we need to learn a new method for calculating the length of curves.
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2.6.2
CHAPTER 2. TOOLS FOR INTEGRATION
Mathematical corner: The length of curves
The length of a curve y = f (x) is calculated in a manner that is very similar to that of the area under the
curve: by breaking down the problem into small pieces, and adding the pieces together!
Consider y = y(x). Suppose we want to calculate its length between the point x = a and x = b.
2.6. INTEGRALS OF FUNCTIONS WITH RADICALS, AND THE LENGTH OF CURVES.
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Once this formula is known, calculating the length of a curve boils down to one differentiation and one
integration.
Example: What is the circumference of a circle?
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2.6.3
CHAPTER 2. TOOLS FOR INTEGRATION
Case Study: The Golden Gate Bridge (part II)
We can now calculate the length of the cables on the main arch of the bridge.
2.6. INTEGRALS OF FUNCTIONS WITH RADICALS, AND THE LENGTH OF CURVES.
2.6.4
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Mathematical corner: Integrating functions with radicals.
In both examples seen above, we ended up having to integrate functions with radicals. Many of the
integrals that you may come across in real life are likely to be on the table provided.
Note that by contrast with rational functions, where one had to be very careful about whether a quadratic
could be factored or not, integral formulas for radical functions are always applicable, e.g.
This is because the method of integration used to obtain this formulas involves “completing the square”,
which can always be done. Let’s just look at this example: