Answer Happy

Determine whether the following statements are true and give an explanation or counter example. Complete parts a through d below. a. If the curve y=f(x) on the interval [a,b] is revolved about the y-axis, the area of the surface generated is ∫f(a)f(b)​2πf(y)1+f′(y)2​dy. A. True. The surface area integral of f(x) when it is rotated about the x-axis on [a,b] is ∫ab​2πf(x)1+f′(x)2​dy. To obtain the surface area of the function when it is rotated about the y-axis, change the limits of integration to f(x) evaluated at each endpoint and integrate with respect to y. This is assuming f(y) is positive on the interval [(a),f(b)]. B. False. To obtain the surface area integral of f(x) when it is rotated about the y-axis on [a,b], the function y=f(x) must be solved for x in terms of y. This yields the function x=g(y). Then t surface area integral becomes ∫f(a)​2πg(y)1+g′(y)2​dy, assuming g(y) is positive on the interval [f(a),f(b)].