Arc Length of Smooth, Planar Curve - AP Calculus BC

1. What is the relationship between curvature and radius of curvature?

   $R = \frac{1}{\kappa}$. Curvature and radius of curvature are reciprocals of each other.

2. What is the curvature of the parametric curve $x(t) = t, y(t) = t^2$?

   $\kappa = \frac{2}{(1 + 4t^2)^{3/2}}$. Applies the parametric curvature formula with $x'(t) = 1, y'(t) = 2t, y''(t) = 2$.

3. Find the total distance traveled by a particle with velocity $v(t) = 3t$ from $t = 0$ to $t = 2$.

   $D = 6$. Distance equals $\int_0^2 3t \, dt = \frac{3t^2}{2}|_0^2 = 6$.

4. Identify the expression for the arc length of a curve $x = g(y)$ from $y = c$ to $y = d$.

   $L = \int_{c}^{d} \sqrt{1 + (g'(y))^2} \, dy$. Similar to $y = f(x)$ formula but with $x$ as a function of $y$.

5. Determine the arc length of $r = 1 + \sin(\theta)$ from $\theta = 0$ to $\theta = \pi$.

   $L = 5.333$ (approximately). The cardioid $r = 1 + \sin(\theta)$ has a complex arc length integral.

6. Find the radius of curvature of $y = \sin(x)$ at $x = 0$.

   $R = 1$. At $x = 0$: $f'(0) = 1, f''(0) = -1$, giving $\kappa = 1$.

7. How do you express arc length in polar coordinates from $\theta = a$ to $\theta = b$?

   $L = \int_{a}^{b} \sqrt{(\frac{dr}{d\theta})^2 + r^2} \, d\theta$. Combines radial and tangential components: $\frac{dr}{d\theta}$ and $r$ respectively.

8. What is the differential arc length $ds$ in polar coordinates $(r, \theta)$?

   $ds = \sqrt{(dr)^2 + (r \, d\theta)^2}$. In polar coordinates, $r \, d\theta$ represents the tangential component of arc length.

9. Determine the arc length of $x(t) = t^2, y(t) = t^3$ from $t = 0$ to $t = 2$.

   $L = \frac{1}{3} (17\sqrt{17} - 1)$. Uses parametric formula with $x'(t) = 2t, y'(t) = 3t^2$, then integrates.

10. Find the arc length for $r = 2\cos(\theta)$ from $\theta = 0$ to $\theta = \frac{\pi}{2}$.

    $L = 2$. For $r = 2\cos(\theta)$, this represents a semicircle with diameter 2.

11. What is the formula for the speed of a particle in terms of its parametric derivatives?

    $v = \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2}$. Speed is the magnitude of the velocity vector in parametric form.

12. Find the arc length of $y = 3x$ from $x = 0$ to $x = 4$.

    $L = 4 \sqrt{10}$. For $y = 3x$, $f'(x) = 3$, so $\sqrt{1 + 9} = \sqrt{10}$ over length 4.

13. What is the formula for the total distance traveled by a particle with position $s(t)$?

    $D = \int_{a}^{b} |v(t)| \, dt$. Integrates the absolute value of velocity to account for direction changes.

14. Compute the radius of curvature for $y = x^3$ at $x = 0$.

    $R = \infty$. At $x = 0$: $f''(0) = 0$, so curvature is 0 and radius is infinite.

15. Calculate the arc length of $y = x^2$ from $x = 0$ to $x = 1$.

    $L = \frac{\sqrt{5} + \ln(1 + \sqrt{5})}{2}$. Uses $f'(x) = 2x$ in the arc length formula and evaluates the resulting integral.

16. Find the total distance traveled by $x(t) = 2t, y(t) = 3t$ from $t = 0$ to $t = 2$.

    $D = 4\sqrt{13}$. Linear motion with constant speed $\sqrt{4 + 9} = \sqrt{13}$ over time interval 2.

17. What is the expression for the curvature $\kappa$ of a curve $y = f(x)$?

    $\kappa = \frac{|f''(x)|}{(1 + (f'(x))^2)^{3/2}}$. Measures how quickly the curve deviates from its tangent line.

18. Calculate the total distance traveled by $x(t) = t, y(t) = t^2$ from $t = 0$ to $t = 3$.

    $D = 5.196$ (approximately). Integrates $\sqrt{1 + 4t^2}$ from 0 to 3 to get the total path length.

19. Calculate the curvature of $r(\theta) = 1 + \cos(\theta)$ at $\theta = 0$.

    $\kappa = \frac{3}{4}$. For cardioid at $\theta = 0$: $r = 2, r' = 0, r'' = -1$.

20. What is the arc length for $y = \sin(x)$ from $x = 0$ to $x = \frac{\pi}{2}$?

    $L = 1.910$ (approximately). The arc length integral for $\sin(x)$ cannot be expressed in elementary functions.

21. What is the formula for the radius of curvature $R$ of a curve $y = f(x)$?

    $R = \frac{1}{\kappa} = \frac{(1 + (f'(x))^2)^{3/2}}{|f''(x)|}$. Radius of curvature is the reciprocal of curvature.

22. What is the curvature of a circle with radius $r$?

    $\kappa = \frac{1}{r}$. For a circle, curvature is the reciprocal of the radius.

23. What is the formula for the arc length of a parametric curve $x(t), y(t)$ from $t = a$ to $t = b$?

    $L = \int_{a}^{b} \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} \, dt$. Combines $x$ and $y$ velocity components using the distance formula in parametric form.

24. Determine the curvature of $y = x^2$ at $x = 1$.

    $\kappa = \frac{2}{(1 + 4)^{3/2}}$. At $x = 1$: $f'(1) = 2, f''(1) = 2$, so $\kappa = \frac{2}{5^{3/2}}$.

25. State the formula for the arc length of a curve $y = f(x)$ from $x = a$ to $x = b$.

    $L = \int_{a}^{b} \sqrt{1 + (f'(x))^2} \, dx$. Uses Pythagorean theorem on small segments with $1 + (f'(x))^2$ under the square root.

26. State the formula for curvature $\kappa$ in parametric form.

    $\kappa = \frac{|x'(t)y''(t) - y'(t)x''(t)|}{((x'(t))^2 + (y'(t))^2)^{3/2}}$. Uses the cross product of velocity and acceleration vectors in the numerator.

27. What is the arc length differential $ds$ in terms of $dx$ and $dy$?

    $ds = \sqrt{(dx)^2 + (dy)^2}$. Represents the infinitesimal arc length element using the Pythagorean theorem.

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