Assume is smooth on [a, b] and partition [a, b] in the usual way. In the kth subinterval [x k-1 , x k ],

Assume ƒ is smooth on [a, b] and partition [a, b] in the usual way. In the kth subinterval [x_k-1, x_k], construct the tangent line to the curve at the midpoint m_k = (x_k-1 + x_k)/2, as in the accompanying figure.

a. Show that

b. Show that the length L_k of the tangent line segment in the kth subinterval is

c. Show that the lateral surface area of the frustum of the cone swept out by the tangent line segment as it revolves about the x-axis is

d. Show that the area of the surface generated by revolving y = ƒ(x) about the x-axis over [a, b] is

= ri f(mk) - f'(m) 2 .. and r = f(m) + '(mk)- 2