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 L 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 $L=\int\limits_1^e\sqrt{1+\left(\frac{dy}{dx}\right)^2}dx=[\frac{dy}{dx}=\frac{1}{e^y}=\frac{1}{x}]=\int\limits_1^e\sqrt{1+\frac{1}{x^2}}dx$

4$L=\int\limits_0^1\sqrt{1+\left(\frac{dx}{dy}\right)^2}dy=[\frac{dx}{dy}=\frac{1}{1/x}=e^y]=\int\limits_0^1\sqrt{1+e^{2y}}dy$