Arc Length — Question 10

Question 10

A curve satisfies $$y'(x)=\sqrt{e^{2x}-1},\qquad0\le x\le\ln3.$$ Find its length. A formula for $y$ is not needed.

Question 10 – Solution

Step 1: Substitute the derivative into the arc-length formula. $$\begin{align*} 1+(y')^2 &=1+\left(\sqrt{e^{2x}-1}\right)^2\\ &=1+e^{2x}-1=e^{2x}. \end{align*}$$ Step 2: Simplify the square root. Since $e^x>0$, $$\sqrt{1+(y')^2}=\sqrt{e^{2x}}=e^x.$$ Step 3: Integrate and evaluate. $$\begin{align*} L&=\int_0^{\ln3}e^x\,dx\\ &=[e^x]_0^{\ln3}\\ &=e^{\ln3}-e^0=3-1=2. \end{align*}$$ $$\boxed{L=2}$$