Question

Surface area of a cone Find the surface area of a cone (excluding the base) with radius 4 and height 8 using integration and a surface area integral.

Accepted Answer

Recall that the lateral surface area of a cone can be found by revolving a line segment (the slant height) around the axis. To use integration, first express the cone's slant height as a function. Set up a coordinate system where the cone's height extends along the x-axis from 0 to 8, and the radius varies linearly from 0 to 4 along this height. Find the equation of the line representing the radius as a function of height x. Since the radius increases linearly from 0 at x=0 to 4 at x=8, the radius function is $$r(x) = \frac{4}{8}x = \frac{x}{2}. $$The surface area generated by revolving a curve $$y = f(x)$$ about the x-axis from $$x=a to$$ $$x=b is $$given by the formula $$S = \int_a^b 2\pi y \sqrt{1 + \left(\frac{dy}{dx}\$$right)^2$$} \, dx. $$Here, $$y = r(x) = \frac{x}{2}. $$Compute the derivative $$\frac{dy}{dx} of $$the radius function: $$\frac{dy}{dx} = \frac{d}{dx} \left( \frac{x}{2} \right) = \frac{1}{2}. $$Substitute $$y$$ and $$\frac{dy}{dx}$$ into the surface area integral to get $$S = \$$int_0$$^8 2\pi \left( \frac{x}{2} \right) \sqrt{1 + \left( \frac{1}{2} \$$right)^2$$} \, dx. $$Simplify the integrand and set up the integral for evaluation. The integral becomes $$S = \$$int_0$$^8 \pi x \sqrt{1 + \frac{1}{4}} \, dx = \$$int_0$$^8 \pi x \sqrt{\frac{5}{4}} \, dx. $$This integral can then be evaluated to find the lateral surface area.

Surface area of a cone Find the surface area of a cone (excluding the base) with radius 4 and height 8 using integration and a surface area integral.

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Key Concepts

Surface Area of a Solid of Revolution

Equation of the Cone's Slant Line

Arc Length Differential (ds)