3.
Changing the parameter of the curve to "x" we get the expression
"y = \\ln x,\\; x \\in [1,e]."
Applying the formula of function graph arc length we get
"L = \\int_1^e \\sqrt{1 + (\\ln'(x))^2} dx = \\int_1^e \\sqrt{1 + \\frac{1}{x^2}} dx \\\\=\n\\int_1^e \\frac{\\sqrt{1+x^2}}{x} dx."
Answer: "L = \\int_1^e \\frac{\\sqrt{1+x^2}}{x} dx."
4.
Similarly, let us change the curve parameter to "y"
"x = e^y,\\;y \\in [0,1]."
Then,
"L = \\int_0^1 \\sqrt{1 + (\\frac{d}{dy}e^y)^2}dy = \\int_0^1 \\sqrt{1 + e^{2y}}dy."
Answer: "L = \\int_0^1 \\sqrt{1 + e^{2y}}dy."
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