Length of a curve given by a function and distance covered by a projectile

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Using differentiation, the slope of a curve can be found and using integration, the area under the curve is found.

How can calculus be used to determine the length of the curve represented by a given function?

If equation of trajectory of the proectile is given by y=f(x) which represents a parabola in a uniform gravitational field, I formulated the integral to find distance covered by the projectile as:
R is the range of the proectile
(R being the range)

I applied it in this question:

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to find the total distance in order to find average speed.

However I did not get the right answer.

What changes need to be made in the integral?
What will be the integral used to find total distance in this question?