91Ó°ÊÓ

The region enclosed by the given curves is rotated about the specified line. Find the volume of the resulting solid. $$x=2 y-y^{2}, x=0 ; \quad \text { about the } y-axis$$

\(13-16=\) The given curve is rotated about the \(y\) -axis. Find the area of the resulting surface. $$y=1-x^{2}, \quad 0 \leqslant x \leqslant 1$$

Show how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it. A 10 -ft chain weighs 25 lb and hangs from a ceiling. Find the work done in lifting the lower end of the chain to the ceiling so that it's level with the upper end.

\(11-20=\) Sketch the region enclosed by the given curves and find its area. $$ y=\cos x, \quad y=2-\cos x, \quad 0 \leqslant x \leqslant 2 \pi $$

\(9-14\) . Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the \(x\) -axis. $$x+y=3, \quad x=4-(y-1)^{2}$$

Find the exact length of the curve. $$y=\frac{1}{4} x^{2}-\frac{1}{2} \ln x, \quad 1 \leqslant x \leqslant 2$$

\(9-14\) . Find the solution of the differential equation that satisfies the given initial condition. $$\frac{d L}{d t}=k L^{2} \ln t, L(1)=-1$$

The region enclosed by the given curves is rotated about the specified line. Find the volume of the resulting solid. $$x-y=1, y=x^{2}-4 x+3 ; \quad \text { about } y=3$$

\(15-20=\) Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis. $$y=x^{4}, y=0, x=1 ; \quad \text { about } x=2$$

\(11-20=\) Sketch the region enclosed by the given curves and find its area. $$ x=2 y^{2}, \quad x=4+y^{2} $$