1. The problem is to find the surface area of an arc, which usually means the surface area of a curved surface generated by rotating an arc around an axis. 2. The formula for the surface area $S$ of a surface of revolution generated by rotating a curve $y=f(x)$ from $x=a$ to $x=b$ about the x-axis is: $$S = 2\pi \int_a^b y \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx$$ 3. Important rules: - $y$ is the distance from the x-axis to the curve (radius of rotation). - $\frac{dy}{dx}$ is the derivative of $y$ with respect to $x$. - The integral sums the surface area of infinitesimal bands along the curve. 4. To find the surface area of an arc, you must: - Express the arc as a function $y=f(x)$ or parametric equations. - Compute $\frac{dy}{dx}$. - Substitute into the formula. - Evaluate the integral over the interval of the arc. 5. Example: If the arc is a semicircle of radius $r$ described by $y=\sqrt{r^2 - x^2}$ from $x=-r$ to $x=r$: - Compute $\frac{dy}{dx} = \frac{-x}{\sqrt{r^2 - x^2}}$. - Then $$\sqrt{1 + \left(\frac{dy}{dx}\right)^2} = \sqrt{1 + \frac{x^2}{r^2 - x^2}} = \sqrt{\frac{r^2}{r^2 - x^2}} = \frac{r}{\sqrt{r^2 - x^2}}$$ - Substitute into surface area formula: $$S = 2\pi \int_{-r}^r \sqrt{r^2 - x^2} \cdot \frac{r}{\sqrt{r^2 - x^2}} \, dx = 2\pi r \int_{-r}^r dx = 2\pi r (2r) = 4\pi r^2$$ 6. This matches the surface area of a sphere of radius $r$, confirming the method. In summary, use the surface of revolution formula with the arc's function and limits to find the surface area.