Answer to Question #100159 in Calculus for Joseph Se

Question

3. Let L be the length of the curve x=e^y from (1,0) to (e,1), integrating with respect to x an integral expression for L is_____.

4. Let be the length of the curve y=ln(x) from (1,0) to (e,1), integrating with respect to y, an integral expression for L is_____.

Answer 1

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

$L = \int_1^e \sqrt{1 + (\ln'(x))^2} dx = \int_1^e \sqrt{1 + \frac{1}{x^2}} dx \\= \int_1^e \frac{\sqrt{1+x^2}}{x} dx.$

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: $L = \int_0^1 \sqrt{1 + e^{2y}}dy.$