AP Calculus BC Unit 8 (BC topic): Arc Length of a Smooth Curve

How long is a curve? Split it into tiny straight pieces of length $\sqrt{(dx)^2+(dy)^2}$ and add them up: the result is an integral. This page covers graphs $y=f(x)$ and $x=g(y)$; Unit 9 extends the same idea to parametric curves.

This page covers only the BC topics of Unit 8. The rest of the unit is the same as AP Calculus AB: AB Unit 8 notes and practice.

BC topics on this page: 8.13 Arc length and distance traveled.

Key formulas and facts

• Length of $y=f(x)$ for $a\le x\le b$: $\displaystyle L=\int_a^b\sqrt{1+\left(f'(x)\right)^2}\,dx$
• Length of $x=g(y)$ for $c\le y\le d$: $\displaystyle L=\int_c^d\sqrt{1+\left(g'(y)\right)^2}\,dy$
• Length function: $L(b)=\int_a^b\sqrt{1+(f'(x))^2}\,dx$ has $L'(b)=\sqrt{1+(f'(b))^2}$
• A curve is never shorter than the straight segment joining its endpoints
• Most arc-length integrals need a calculator; exact answers come when $1+(f')^2$ is a perfect square

Worked examples

• $y=x^{3/2}$ on $[0,4]$: $\int_0^4\sqrt{1+\frac94x}\,dx=\frac{8}{27}\left(10^{3/2}-1\right)$
• Perfect square: $y=\frac{x^3}{6}+\frac1{2x}$ on $[1,2]$ has $1+(y')^2=\left(\frac{x^2}{2}+\frac1{2x^2}\right)^2$, so $L=\int_1^2\left(\frac{x^2}{2}+\frac{1}{2x^2}\right)dx=\frac{17}{12}$
• Calculator: $y=e^x$ on $[0,1]$ has $L=\int_0^1\sqrt{1+e^{2x}}\,dx\approx2.003$
• Straight line check: $y=2x$ on $[0,3]$ gives $\int_0^3\sqrt5\,dx=3\sqrt5$, the distance from $(0,0)$ to $(3,6)$

Common traps

• Forgetting to square the derivative, or writing $\sqrt{1+f'(x)}$.
• Writing $\sqrt{1+(f'(x))^2}=1+f'(x)$: a square root of a sum is not the sum of square roots.
• Using the $x$-formula for a curve given as $x=g(y)$, or mixing $dx$ limits with a $dy$ integrand.
• Rounding the integrand before integrating on the calculator: keep full precision and round only the final answer to three decimal places.

Practice questions

8 multiple-choice questions and 1 free-response question. Work each one out before you open the answer. In a free-response scoring guide, each line is worth the points it states. A free account opens the worked answers and scoring guides of 20 questions a day; Pro opens every one.

Question 1 · Multiple choice

What is the length of the graph of $y=\frac23x^{3/2}$ from $x=0$ to $x=3$?

1. $\frac{16}{3}$
2. $2 \sqrt{3}$
3. $\frac{14}{3}$
4. $\frac{7}{3}$

Question 2 · Multiple choice

What is the length of the curve $y=\dfrac{x^2}{8}-\ln x$ for $1\le x\le2$?

1. $- \frac{3}{8} + \ln{\left(2 \right)}$
2. $\frac{3}{8} + \ln{\left(2 \right)}$
3. $\frac{3}{8}$
4. $\ln{\left(2 \right)} + \frac{3}{4}$

Question 3 · Multiple choice

Which of the following integrals gives the length of the graph of $y=\ln x$ from $x=1$ to $x=3$?

1. $\displaystyle\int_1^3\sqrt{1+(\ln x)^2}\,dx$
2. $\displaystyle\int_1^3\sqrt{1+\frac1x}\,dx$
3. $\displaystyle\int_1^3\left(1+\frac{1}{x^2}\right)dx$
4. $\displaystyle\int_1^3\sqrt{1+\frac{1}{x^2}}\,dx$

Question 4 · Multiple choice

What is the length of the graph of $y=x^3$ from $x=0$ to $x=1$?

1. $1.414$
2. $1.25$
3. $1.548$
4. $2.0$

Question 5 · Multiple choice

The integral $\displaystyle\int_1^4\sqrt{1+9x}\,dx$ gives the length of the graph of which function on $1\le x\le4$?

1. $y=2x^{3/2}$
2. $y=3\sqrt x$
3. $y=x^{3/2}$
4. $y=\frac92x^2$

Question 6 · Multiple choice

What is the length of the curve $x=\dfrac{y^2}{2}$ from $y=0$ to $y=1$?

1. $\sqrt{2}$
2. $\ln{\left(1 + \sqrt{2} \right)} + \sqrt{2}$
3. $\frac{\sqrt{5}}{2}$
4. $\frac{\ln{\left(1 + \sqrt{2} \right)}}{2} + \frac{\sqrt{2}}{2}$

Question 7 · Multiple choice

What is the length of the graph of $y=3x+1$ for $0\le x\le4$?

1. $12$
2. $4 \sqrt{10}$
3. $16$
4. $2 \sqrt{10}$

Question 8 · Multiple choice

The figure shows the graph of $y=\cos x$ for $0\le x\le\pi$. What is the length of this graph?

1. $3.142$
2. $2.0$
3. $3.724$
4. $3.82$

Question 9 · Free response (9 points)

Let $f(x)=\frac13\left(x^2+2\right)^{3/2}$ for $x\ge0$, and let $L(b)$ be the length of the graph of $f$ from $x=0$ to $x=b$.

1. (a) Find $f'(x)$. [1 point]
2. (b) Show that $1+\left(f'(x)\right)^2=\left(x^2+1\right)^2$, and find $L(3)$. [3 points]
3. (c) Find $L'(2)$, and explain what it tells you about the graph of $f$. [2 points]
4. (d) Write $L(b)$ as an expression in $b$, and find the value of $b$ for which $L(b)=\frac43$. [3 points]