Welcome back to class! In previous sections, we used integration to find the area under a curve and the volume of solids of revolution. Now, we are asking a different question: How long is the curve itself? In Section 8-1, we explore the concept of Arc Length.
The Concept
To find the length of a curve, we can approximate it by dividing the curve into small line segments. As the number of segments approaches infinity (and their length approaches zero), the sum of these segments becomes the exact length of the curve. This derivation leads us directly to the Arc Length Formula, which is essentially an integral version of the Pythagorean Theorem/Distance Formula.
The Formulas
Depending on whether the function is easier to integrate with respect to $x$ or $y$, we have two variations of the formula. If $f'$ is continuous on $[a, b]$, the length $L$ of the curve $y = f(x)$ is:
$$L = \int_{a}^{b} \sqrt{1 + \left[f'(x)\right]^2} \, dx = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx$$However, if the curve is defined as $x = g(y)$ and we are integrating from $c$ to $d$ on the y-axis, we use:
$$L = \int_{c}^{d} \sqrt{1 + \left[g'(y)\right]^2} \, dy = \int_{c}^{d} \sqrt{1 + \left(\frac{dx}{dy}\right)^2} \, dy$$Key Examples from Class Notes
In the attached notes, we walk through several specific examples that highlight different integration techniques required for these problems.
- Semicubical Parabola ($y^2 = x^3$): We solved for arc length between $(1,1)$ and $(4,8)$. This example required us to find the derivative $\frac{dy}{dx} = \frac{3}{2}x^{1/2}$, square it to get $\frac{9}{4}x$, and then use u-substitution to integrate $\int \sqrt{1 + \frac{9}{4}x} \, dx$.
- Integration Complexity ($ sec^3 \theta$): As noted in the handwritten slides (specifically regarding the parabola $y^2 = x$), arc length problems often result in integrals involving square roots. This frequently requires Trigonometric Substitution. You will often encounter the integral of $\sec^3 \theta$, which requires integration by parts. It is a good idea to review this specific integral, as it appears often in this chapter!
The Arc Length Function
We also defined the Arc Length Function $s(x)$, which measures the distance along a curve from a starting point $a$ to a variable point $x$.
$$s(x) = \int_{a}^{x} \sqrt{1 + [f'(t)]^2} \, dt$$Professor's Tip: In Example 4 of the notes (involving the natural log function), notice the algebraic magic that happens under the radical. Often, textbook problems are designed so that $1 + (f'(x))^2$ simplifies into a perfect square, allowing the square root to cancel out. Watch for this pattern!
Be sure to download the PowerPoint and the handwritten Class Notes below to see the step-by-step derivations and the full solution for the trigonometric substitution example.