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A 3 lb bucket containing 20 lb of water is hanging at the end of a 20 ft rope that weighs 4 oz/ft. The other end of the rope is attached to a pulley. How much work is required to wind the length of rope onto the pulley, assuming that the rope is wound onto the pulley at a rate of \(2 \mathrm{ft} / \mathrm{s}\) and that as the bucket is being lifted, water leaks from the bucket at a rate of \(0.5 \mathrm{lb} / \mathrm{s} ?\)

Use symmetry considerations to argue that the centroid of an ellipse lies at the intersection of the major and minor axes of the ellipse.

Use a CAS to find the volume of the solid generated when the region enclosed by \(y=\cos x, y=0,\) and \(x=0\) for \(0 \leq x \leq \pi / 2\) is revolved about the \(y\) -axis.

A basketball player makes a successful shot from the free throw line. Suppose that the path of the ball from the moment of release to the moment it enters the hoop is described by $$ y=2.15+2.09 x-0.41 x^{2}, \quad 0 \leq x \leq 4.6 where \(x\) is the horizontal distance (in meters) from the point of release, and \(y\) is the vertical distance (in meters) above the floor. Use a CAS or a scientific calculator with a numerical integration capability to approximate the distance the ball travels from the moment it is released to the moment it enters the hoop. Round your answer to two decimal places. $$

Find the volume of the solid that results when the region enclosed by the given curves is revolved about the \(y\) -axis. $$ y=x^{2}, x=y^{2} $$

Find \(d y / d x\) $$ y=\cosh ^{-1}\left(\sinh ^{-1} x\right) $$

Use a graphing utility, where helpful, to find the area of the region enclosed by the curves. $$ y=x^{3}-4 x, y=0 $$

Use a graphing utility, where helpful, to find the area of the region enclosed by the curves. $$ x=y^{3}-y, x=0 $$

A rocket weighing 3 tons is filled with 40 tons of liquid fuel. In the initial part of the flight, fuel is burned off at a constant rate of 2 tons per \(1000 \mathrm{ft}\) of vertical height. How much work in foot-tons (ft-ton) is done lifting the rocket \(3000 \mathrm{ft}\) ?

Find \(d y / d x\) $$ y=\frac{1}{\tanh ^{-1} x} $$