91Ó°ÊÓ

Given the region bounded by the graphs of \(y=\ln x, y=0\), and \(x=e\), find (a) the area of the region. (b) the volume of the solid generated by revolving the region about the \(x\) -axis. (c) the volume of the solid generated by revolving the region about the \(y\) -axis. (d) the centroid of the region.

The inner product of two functions \(f\) and \(g\) on \([a, b]\) is given by \(\langle f, g\rangle=\int_{a}^{b} f(x) g(x) d x\). Two distinct functions \(f\) and \(g\) are said to be orthogonal if \(\langle f, g\rangle=0\). Show that the following set of functions is orthogonal on \([-\pi, \pi]\). \(\\{\sin x, \sin 2 x, \sin 3 x, \ldots, \cos x, \cos 2 x, \cos 3 x, \ldots\\}\)

Prove the following generalization of the Mean Value Theorem. If \(f\) is twice differentiable on the closed interval \([a, b]\), then $$ f(b)-f(a)=f^{\prime}(a)(b-a)-\int_{a}^{b} f^{\prime \prime}(t)(t-b) d t $$