AP Calculus BC / Unit 8: Applications of Integration
8.13 The Arc Length of a Smooth, Planar Curve and Distance Traveled
Notes
$$L=\int_a^b\sqrt{1+[f'(x)]^2}\,dx$$
- Most AP arc length integrals are done on a calculator.
Practice
Arc length of $\displaystyle y=\frac{2}{3}x^{3/2}$ from 0 to 3:
$\displaystyle \int_0^3\sqrt{1+x}\,dx=\frac{2}{3}(8-1)$.
The arc length of $y=3x+1$ from 0 to 4 is:
$\displaystyle \int_0^4\sqrt{10}\,dx$.
Which integral gives the length of $y=x^2$ from 0 to 1?
$f'=2x$, squared is $4x^2$.
Ready for more? Take the Unit 8 test.