91Ó°ÊÓ

Evaluate \(\int \sin x \cos x d x\) by four methods: $$\begin{array}{l}{\text { (a) the substitution } u=\cos x} \\ {\text { (b) the substitution } u=\sin x} \\ {\text { (c) the identity } \sin 2 x=2 \sin x \cos x} \\ {\text { (d) integration by parts }}\end{array}$$ Explain the different appearances of the answers.

\(55-56\) Use a graph to find approximate \(x\) -coordinates of the points of intersection of the given curves. Then find (approxi- mately) the area of the region bounded by the curves. $$y=\arctan 3 x, \quad y=\frac{1}{2} x$$

\(57-60\) Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis. $$y=\cos (\pi x / 2), y=0,0 \leqslant x \leqslant 1 ; \quad \text about\quad the\quad y -axis $$

The German mathematician Karl Weierstrass \((1815-1897)\) noticed that the substitution \(t=\tan (x / 2)\) will convert any rational function of \(\sin x\) and \(\cos x\) into an ordinary rational function of \(t .\) (a) If \(t=\tan (x / 2),-\pi

Find the area of the region bounded by the given curves. $$y=\sin ^{2} x, \quad y=\cos ^{2} x, \quad-\pi / 4 \leqslant x \leqslant \pi / 4$$

\(57-59\) Find the values of \(p\) for which the integral converges and evaluate the integral for those values of \(p .\) $$\int_{0}^{1} \frac{1}{x^{p}} d x$$

Find the area of the region bounded by the given curves. $$y=\sin ^{3} x, \quad y=\cos ^{3} x, \quad \pi / 4 \leqslant x \leqslant 5 \pi / 4$$

\(1-80\) Evaluate the integral. $$\int \frac{x \ln x}{\sqrt{x^{2}-1}} d x$$

\(57-60\) Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis. $$y=e^{x}, y=e^{-x}, x=1 ; \quad \text { about the } y -axis $$

\(57-60\) Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis. $$y=e^{-x}, y=0, x=-1, x=0 ; \quad \text { about } x=1$$