Arc Length — Question 2

Question 2

For $0\le x\le2$, compare $$C_1:y=0,\qquad C_2:y=\frac23x^{3/2}.$$ Find $L_2-L_1$.

Question 2 – Solution

Step 1: Find the length of $C_1$. Since $y'=0$, $$L_1=\int_0^2\sqrt{1+0^2}\,dx=\int_0^2 1\,dx=2.$$ Step 2: Differentiate $C_2$. $$y'=\frac23\cdot\frac32x^{1/2}=\sqrt{x}.$$ Step 3: Find the length of $C_2$. $$\begin{align*} L_2&=\int_0^2\sqrt{1+(\sqrt{x})^2}\,dx\\ &=\int_0^2\sqrt{1+x}\,dx\\ &=\left[\frac23(1+x)^{3/2}\right]_0^2\\ &=\frac23(3\sqrt3-1)=2\sqrt3-\frac23. \end{align*}$$ Step 4: Subtract. $$L_2-L_1=2\sqrt3-\frac23-2=2\sqrt3-\frac83.$$ $$\boxed{L_2-L_1=2\sqrt3-\frac83}$$