Write the equation of the surface of revolution formed by revolving the profile curve \( y=2 x+3 \) around the \( x \)-axis.

To find the equation of the surface generated by revolving the curve \( y = 2x + 3 \) around the \( x \)-axis, we can use the formula for the surface of revolution. The general equation for the surface formed by revolving a function \( y = f(x) \) about the \( x \)-axis is given by: \[ S = 2\pi \int_a^b y \, \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \] For our function \( y = 2x + 3 \), first, we need to compute the derivative: \[ \frac{dy}{dx} = 2 \] Then, substituting \( y \) and \( \frac{dy}{dx} \) into the surface area formula: \[ \sqrt{1 + \left( \frac{dy}{dx} \right)^2} = \sqrt{1 + 2^2} = \sqrt{5} \] The surface of revolution equation can thus be represented parametrically for any point on the surface as: \[ x = x, \quad y = 2x + 3 \cos(\theta), \quad z = 3 \sin(\theta) \] Where the angle \( \theta \) varies from \( 0 \) to \( 2\pi \) to complete the revolution. Now, think of it like a party hat. If you take that linear profile and spin it around, you'll create a funky, fun surface that expands and narrows, just like our hat!