Arc Length & Surface Area

Cheatsheet Content

### Arc Length of a Curve The arc length $L$ of a smooth curve $y = f(x)$ from $x=a$ to $x=b$ is given by the integral: $$L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} dx$$ If the curve is defined parametrically by $x=f(t)$ and $y=g(t)$ from $t=a$ to $t=b$, the arc length is: $$L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} dt$$ ### Surface Area of Revolution The surface area $S$ generated by revolving a curve $y = f(x)$ (where $f(x) \ge 0$) from $x=a$ to $x=b$ about the x-axis is: $$S = \int_a^b 2\pi y \sqrt{1 + \left(\frac{dy}{dx}\right)^2} dx$$ If the curve is revolved about the y-axis (and $x=g(y) \ge 0$), the surface area is: $$S = \int_c^d 2\pi x \sqrt{1 + \left(\frac{dx}{dy}\right)^2} dy$$ For parametric curves $x=f(t)$, $y=g(t)$ revolved about the x-axis ($g(t) \ge 0$): $$S = \int_a^b 2\pi y \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} dt$$ ### Example: Arc Length Calculation **Problem:** Find the arc length of $y = 4x - 4$ on $0 \le x \le 4$. **Solution Steps:** 1. **Find the derivative:** $\frac{dy}{dx} = 4$ 2. **Square the derivative:** $\left(\frac{dy}{dx}\right)^2 = 4^2 = 16$ 3. **Set up the integral:** $$L = \int_0^4 \sqrt{1 + 16} dx = \int_0^4 \sqrt{17} dx$$ 4. **Evaluate the integral:** $$L = \left[\sqrt{17}x\right]_0^4 = 4\sqrt{17}$$ ### Example: Surface Area of Revolution **Problem:** Find the surface area of revolution about the x-axis of $y = 7x + 2$ over the interval $0 \le x \le 2$. **Solution Steps:** 1. **Find the derivative:** $\frac{dy}{dx} = 7$ 2. **Square the derivative:** $\left(\frac{dy}{dx}\right)^2 = 7^2 = 49$ 3. **Set up the integral:** $$S = \int_0^2 2\pi (7x+2) \sqrt{1 + 49} dx = \int_0^2 2\pi (7x+2) \sqrt{50} dx$$ $$S = 2\pi \cdot 5\sqrt{2} \int_0^2 (7x+2) dx = 10\pi\sqrt{2} \int_0^2 (7x+2) dx$$ 4. **Evaluate the integral:** $$S = 10\pi\sqrt{2} \left[\frac{7}{2}x^2 + 2x\right]_0^2$$ $$S = 10\pi\sqrt{2} \left(\left(\frac{7}{2}(2^2) + 2(2)\right) - (0)\right)$$ $$S = 10\pi\sqrt{2} \left(\frac{7}{2}(4) + 4\right) = 10\pi\sqrt{2} (14 + 4)$$ $$S = 10\pi\sqrt{2} (18) = 180\pi\sqrt{2} \approx 799.72$$ ### Common Formulas and Simplifications - **If $y = x^k$:** $\frac{dy}{dx} = kx^{k-1}$ - **If $y = e^{ax}$:** $\frac{dy}{dx} = ae^{ax}$ - **If $y = \sin(ax)$:** $\frac{dy}{dx} = a\cos(ax)$ - **Often, $\sqrt{1 + (dy/dx)^2}$ needs simplification.** Look for perfect squares, e.g., if $1 + (dy/dx)^2 = (A+B)^2$. - Example: If $1 + (dy/dx)^2 = \left(\frac{e^x - e^{-x}}{2}\right)^2 + 1 = \left(\frac{e^x + e^{-x}}{2}\right)^2$ ### Tips and Tricks - **Visualize:** Sketch the curve to understand the problem. - **Check Limits:** Ensure the integration limits correspond to the given interval for $x$, $y$, or $t$. - **Algebra is Key:** Many problems involve significant algebraic manipulation to simplify the integrand. - **U-Substitution:** Frequently used to solve the resulting integrals. - **Tables/Technology:** For complex integrals, use integral tables or computational tools as needed.