There are quite simple, intuitive and straightforward expressions for evaluating the area or volume of a figure. But why is the expression for the length of a curve so complicated?
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The length of a curve doesn't transform in a nice way under scaling of one of the variables the same way that the area of a 2d figure or the volume of a 3d figure does. That puts a hard lower bound on how complicated it has to be.
Obviously this depends on the shape, as you implied. I would say for a circle the circumference formula is simpler.
Also the length of the curve is a volume in some sense, its just 1 dimensional.
Usually in calculus we are calculating the area relative to the the x-axis, and your formula is in terms of $x$, so that makes things simpler. I imagine that if the curve was parameterized in terms of its arc length finding its length would be quite trivial, and finding the area between the curve and the x-axis would seem very complicated.