Arc Length & Surface Area

Part 1 of 7 — Arc Length in Rectangular Form

Arc Length Formula ($y = f(x)$)

Arc Length 🎯

Key Takeaways — Part 1

Arc length = $\int \sqrt{1 + (dy/dx)^2}\,dx$.

Arc Length

Part 2 of 7 — Parametric Arc Length

Arc Length (Parametric)

Parametric/Polar Arc Length 🎯

Key Takeaways — Part 2

Polar arc length: $\sqrt{r^2 + (dr/d\theta)^2}\,d\theta$.

Arc Length & Surface Area

Part 3 of 7 — Surface Area of Revolution

Around the $x$-axis

$S = 2piint_a^b ysqrt{1 + (y')^2},dx$

Around the $y$-axis

$S = 2piint_a^b xsqrt{1 + (y')^2},dx$

Surface Area 🎯

Key Takeaways — Part 3

$S = 2\pi\int r \cdot ds$ where $r$ is the radius and $ds$ is the arc length element.

Arc Length

Part 4 of 7 — Speed and Arc Length Connection

Speed Function

ext{Speed}(t) = sqrt{[x'(t)]^2 + [y'(t)]^2} = rac{ds}{dt}

So arc length = $int$ speed $cdot dt$

This connects parametric arc length to physics: distance = integral of speed.

Speed Connection 🎯

Key Takeaways — Part 4

Arc length is the integral of the speed function.

Arc Length

Part 5 of 7 — Arc Length with $x = g(y)$

When $x$ is a Function of $y$

$x = g(y)$ Form 🎯

Key Takeaways — Part 5

Sometimes integrating with respect to $y$ gives a simpler integral.

Arc Length

Part 6 of 7 — Practice Workshop

Mixed Practice 🎯

Workshop Complete!

Arc Length & Surface Area — Review

Part 7 of 7 — Final Assessment

Final Assessment 🎯

Arc Length & Surface Area — Complete! ✅