Length of a Curve (Distance Traveled Along a Curve)
If $f^{\prime}$ is continuous on the closed interval $[a, b]$, then the length of the curve $y=f(x)$ from $x=a$ to $x=b$ is
$L=\int_a^b \sqrt{1+\left[\frac{d y}{d x}\right]^2} d x=\int_a^b \sqrt{1+\left[f^{\prime}(x)\right]^2} d x$
If $g^{\prime}$ is continuous on the closed interval $[c, d]$, then the length of the curve $x=g(y)$ from $y=c$ to $y=d$ is
$
L=\int_c^d \sqrt{1+\left[\frac{d x}{d y}\right]^2} d y=\int_c^d \sqrt{1+\left[g^{\prime}(y)\right]^2} d y
$
Example 1
- Find the length of the curve $y=2 x^{3 / 2}+1$, from $x=1$ to $x=3$.
▶️Answer/Explanation
Solution
$\begin{aligned} & \frac{d y}{d x}=2 \cdot \frac{3}{2} x^{1 / 2}=3 x^{1 / 2} \\ &\left(\frac{d y}{d x}\right)^2=\left(3 x^{1 / 2}\right)^2=9 x \\ & L=\int_1^3 \sqrt{1+\left[\frac{d y}{d x}\right]^2} d x=\int_1^3 \sqrt{1+9 x} d x \\ &=\frac{2}{3} \cdot \frac{1}{9}\left[(1+9 x)^{3 / 2}\right]_1^3 \approx 8.633\end{aligned}$
Example2
- Let $R$ be the region bounded by the $y$-axis and the graphs of $y=x^2$ and $y=x+2$.
Find the perimeter of the region $R$.
▶️Answer/Explanation
Solution
The $y$-intercept of the line is $A(0,2)$. We can find the point of intersection of the two graphs by solving $y=x^2$ and $y=x+2$ simultaneously for $x$. The point of intersection is $B(2,4)$.
The perimeter of the region $R$
$=O A+A B+$ length of the curve from $O$ to $B$
$=2+\sqrt{(4-2)^2+(2-0)^2}+\int_0^2 \sqrt{1+(2 x)^2} d x \quad \frac{d y}{d x}=2 x$
$
\begin{aligned}
& =2+2 \sqrt{2}+4.647 \\
& =9.475
\end{aligned}
$
Use a graphing calculator to find
$
\text { the value of } \int_0^2 \sqrt{1+(2 x)^2} d x \text {. }
$
Example 3
- What is the length of the curve of $y=\frac{1}{3}\left(x^2+2\right)^{3 / 2}$ from $x=1$ to $x=2$ ?
(A) $\frac{8}{3}$
(B) $\frac{10}{3}$
(C) 4
(D) $\frac{14}{3}$
▶️Answer/Explanation
Ans:B
Example 4
- Which of the following integrals gives the length of the graph of $y=\ln (\sin x)$ between $x=\frac{\pi}{3}$ to $x=\frac{2 \pi}{3}$ ?
(A) $\int_{\pi / 3}^{2 \pi / 3} \csc ^2 x d x$
(B) $\int_{\pi / 3}^{2 \pi / 3} \sqrt{1+\cot x} d x$
(C) $\int_{\pi / 3}^{2 \pi / 3} \csc x d x$
(D) $\int_{\pi / 3}^{2 \pi / 3} \sqrt{1+\csc ^2 x} d x$
▶️Answer/Explanation
Ans:C
Example 5
- Which of the following integrals gives the length of the graph of $y=\frac{1}{3} x^{3 / 2}-x^{1 / 2}$ between $x=1$ to $x=4 ?$
(A) $\frac{1}{2} \int_1^4\left(\sqrt{x}+\frac{1}{\sqrt{x}}\right) d x$
(B) $\frac{1}{2} \int_1^4\left(\sqrt{x}-\frac{1}{\sqrt{x}}\right) d x$
(C) $\frac{1}{2} \int_1^4\left(1+\sqrt{x}+\frac{1}{\sqrt{x}}\right) d x$
(D) $\frac{1}{2} \int_1^4\left(1+\sqrt{x}-\frac{1}{\sqrt{x}}\right) d x$
▶️Answer/Explanation
Ans:A
Example6
- If the length of a curve from $(0,-3)$ to $(3,3)$ is given by $\int_0^3 \sqrt{1+\left(x^2-1\right)^2} d x$, which of the following could be an equation for this curve?
(A) $y=\frac{x^3}{3}-\frac{x}{3}-3$
(B) $y=\frac{x^3}{3}-3 x-3$
(C) $y=\frac{x^3}{3}-x-3$
(D) $y=\frac{x^3}{3}+x-3$
▶️Answer/Explanation
Ans:D
Example7
- If $F(x)=\int_1^{x^2} \sqrt{t+1} d t$, what is the length of the curve of from $x=1$ to $x=2$ ?
(A) $\frac{8}{3}$
(B) $\frac{10}{3}$
(C) $\frac{15}{3}$
(D) $\frac{17}{3}$
▶️Answer/Explanation
Ans:C