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Magazine Problem 3
A particle moves a distance of \(100 \mathrm{ft}\) along a straight line. As it moves, it is acted upon by a constant force of magnitude \(5 \mathrm{lb}\) in a direction opposite to that of the motion. What is the work done by the force?
Problem 4
Use the method of cylindrical shells to find the volume of the solid generated by revolving the region about the indicated axis or line.
Problem 6
Use the method of cylindrical shells to find the volume of the solid generated by revolving the region about the indicated axis or line.
Problem 10
Use the method of cylindrical shells to find the volume of the solid generated by revolving the region bounded by the graphs of the equations and/or inequalities about the indicated axis. Sketch the region and a representative rectangle. \(y=\sqrt{x-1}, \quad y=0, \quad x=5 ;\) the \(y\) -axis
Problem 12
Use the method of cylindrical shells to find the volume of the solid generated by revolving the region bounded by the graphs of the equations and/or inequalities about the indicated axis. Sketch the region and a representative rectangle. \(y=e^{-x^{2}}, \quad y=0, \quad x=0, \quad x=1 ; \quad\) the \(y\) -axis
Problem 13
Find the centroid of the region bounded by the graphs of the given equations. $$ y=|x| \sqrt{1-x^{2}}, \quad y=0, \quad x=-1, \quad x=1 $$
Problem 15
Use the method of cylindrical shells to find the volume of the solid generated by revolving the region bounded by the graphs of the equations and/or inequalities about the indicated axis. Sketch the region and a representative rectangle. \(x=\sqrt{9-y^{2}}, \quad x=0, \quad y=0 ;\) the \(x\) -axis
Problem 16
Prove the identity. \(\tanh (x+y)=\frac{\tanh x+\tanh y}{1+\tanh x \tanh y}\)
Problem 18
An aquarium has the shape of a rectangular tank of length \(4 \mathrm{ft}\), width \(2 \mathrm{ft}\), and height \(3 \mathrm{ft}\). If the tank is filled with water weighing \(62.4 \mathrm{lb} / \mathrm{ft}^{3}\), find the work required to empty the tank by pumping the water over the top of the tank.
Problem 24
Find the centroid of the region bounded by the graphs of the given equations. $$ y=-x^{2}+3, \quad y=x^{2}-2 x-1 $$
