2. Surfaces of E- a1 Gold Coat Previously, we saw two solids of revolution (a circular cylinder and a sphere) such that when sliced into pieces of equal width, each piece has equal surface area. In other words, the surface area of each piece depends only on the width of the piece. This problem seeks to find descriptions of other possible surfaces of revolution that have this property. Let y : f(t) be a curve above the t-axis defined for all t-values. Rotate this curve around the t-axis to create a surface. Consider the part of the solid in the interval [a, a+h]. We want the solid to meet the condition that the amount of surface area contained in the slice [a, a+h] is proportional to only h (the width of the slice) and doesn't depend on a (the location of the slice). (T he function shown in the image below does not satisfy this property because the blue slice on the left has less surface area than that green slice on the right.) y ml In symbols, we seek to solve the integral equation a+h kh=/ 21rf(t)'/l +(f'(r))2 dz where k>0 is the proportionality constant. a) Differentiate both sides of this equation with respect to h. You'll use the Fundamental Theorem on the right side. You should see terms like f(a+h) and f'(a+h) in your result from above. To make the computations simpler, substitute x=a+h. You can think of this as shifting the section ofyour surface to the origin. Now you have a separable differential equation. b) One function we know has this property is the constant function f(x) = b. Find the value of k so that f(x) = b is a solution. How does this value of k relate to your answer from the previous WH question? Based on your answer from the previous WH, what would you guess is the value of k for which the function '3 f(x) = V R _ x isa solution to the differential equation? C) We will search for other possible surfaces. In your separable differential equation, it may be helpful to write f(x) = y and f'(x) = y' for simplicity. Separate and solve this differential equation for y. Your result will depend on k and C, an integration constant. What kind of curve is this