91Ó°ÊÓ

Find the coordinates of the centroids of the given figures. In Exercises \(11-22,\) each region is covered by a thin, flat plate. The region bounded by \(y=x^{3}, x=2,\) and the \(x\) -axis

A chain is being unwound from a winch. The force of gravity on it is \(12.0 \mathrm{N} / \mathrm{m}\). When \(20 \mathrm{m}\) have been unwound, how much work is done by gravity in unwinding another \(30 \mathrm{m} ?\)

Find the areas bounded by the indicated curves. $$x=y^{2}-4 y, x=0$$

Find the volume generated by revolving the regions bounded by the given curves about the \(x\) -axis. Use the indicated method in each case.$$\left.y=6-x-x^{2}, x=0, y=0 \quad \text { (quadrant } I\right),(disks)$$.

Find the indicated moment of inertia or radius of gyration. Find the radius of gyration of a plate covering the region bounded by \(y^{2}=x^{3}, y=8,\) and the \(y\) -axis with respect to the \(y\) -axis.

A catapult can launch a plane from the deck of an aircraft carrier from 0 to \(260 \mathrm{km} / \mathrm{h}\) in \(2.0 \mathrm{s}\). How many \(g^{\prime}\) s is the average acceleration for such a launch? \(\left(1 g=9.8 \mathrm{m} / \mathrm{s}^{2}\right)\)

Find the areas bounded by the indicated curves. $$y=6-3 x, x=0, y=0, y=3$$

Find the volume generated by revolving the regions bounded by the given curves about the \(x\) -axis. Use the indicated method in each case.$$x=4 y-y^{2}-3, x=0 \quad (shells)$$

Find the coordinates of the centroids of the given figures. In Exercises \(11-22,\) each region is covered by a thin, flat plate. The region bounded by \(y=4 x^{2}\) and \(y=2 x^{3}\)

Find the coordinates of the centroids of the given figures. In Exercises \(11-22,\) each region is covered by a thin, flat plate. The region bounded by \(y^{2}=x, y=2,\) and \(x=0\)