4. Find the area of the surface generated by revolving the cardioid r = 1 + cosθ around the x-axis. __(d)__

2. (10 分) Let C be a positively oriented, piecewise-smooth, simple closed curve in the plane and let D be the region bounded by C. If f(x, y) is a scalar function with continuous first and second partial derivatives in D, derive the following formula: ∬D ( fxx + fyy )dxdy =where = ∇f⋅N is the normal derivative, N is an outer unit normal, s is the arc-length.