The graph of the solution is shown above.  Click on the graph to see an animation of the direction field vectors moving across the screen for increasing values of x along with an animated solution point.  Solutions corresponding to initial conditions y(0) = -1, -0.4, 0.4, 1 are shown below along with the direction field.  Click on the picture to see an enlargement.

The population of a country is growing at a rate that is proportional to the population of the country.  The population in 1990 was 20 million and in 2000 the population was 22 million.  Estimate the population in 2020.

Maple Worksheet

The picture at the right shows the graph of the curve being revolved about the x-axis.

DPGraph Picture 1--Use the scrollbar and activate "b" to revolve the region about the x-axis and form the solid.

DPGraph Picture 2--Use the scrollbar and activate "b" to revolve the region about the x-axis and form the solid.  This time the solid will be transparent.  Use the scrollbar and activate "a" to look at cross section disks.

DPGraph Picture 3--Use the scrollbar and the x slice feature to look at cross sections.

DPGraph Picture 4--Use the scrollbar and activate "a" to look at cross sections between two parallel planes that are perpendicular to the x-axis.  

DPGraph Picture 5--Use the scrollbar and activate "a" to look at lateral surfaces of thick cross section disks.  

DPGraph Picture 6--Use the scrollbar and activate "a" to look at cross section disks only (no solid).  You can activate "b" to draw in the solid.

DPGraph Picture 7--The same as DPGraph Picture 6 except that "b" changes with time to produce an animated drawing of the solid.

DPGraph Picture of the solid of revolution--Use the scrollbar to vary the value of "a" and see various "cylindrical shells".  Each "shell" has a "thickness" of 1/2 unit.

DPGraph Picture2--Use the scrollbar to vary the value of "a" again to see the "shells".  In this case the "shells" are thin.

Find the volume of the solid whose base lies in the xy-coordinate plane and is bounded by the graphs of y = x² and y = 4 and vertical cross sections perpendicular to the y-axis are squares.

DPGraph Picture 2--Use the scrollbar to activate "a" to look at individual cross section squares inside a transparent image of the solid.

DPGraph Picture 3--This is an automated animation of DPGraph Picture 2.

DPGraph Picture 4--This is similar to DPGraph Picture 2 except that one side of the solid is left "open" and the solid is not transparent.  Use the scrollbar to activate "a" again.

Work Pumping Water

The water in a large horse watering trough weighs 62.4 pounds per cubic foot.  The ends of the trough are isosceles triangles with a base of length 10 feet, equal length sides 13 feet, height 12 feet, with the base up as shown in the picture.  The trough is 30 feet long and held in an upright position by supports on the sides.  The trough is completely filled with water.  How much work is done in pumping the water over the edge of the trough to completely empty it?

Solution Below

A force of 100 pounds will stretch a spring 2 feet beyond its equilibrium length of 5 feet.  Find the work done in stretching the spring from a length of 5 feet to a length of 8 feet.  Click here to see an animation and click here to see an animation with scales.

100 = 2k so the spring constant is 50. The work done would be

How much additional work would be done in stretching the spring two more feet (assuming we are still within the elastic limits of the spring and Hooke's Law holds)?

Center of Mass

Find the center of mass of a planar lamina whose density is 3 units/square unit and whose boundaries are formed by the graphs of the functions given below.

Center of Mass

Find the center of mass of a planar lamina whose density is 2 units/square unit and whose boundaries are formed by the graphs of the functions given below.

Extra Credit:  Find the mass of a plate with the same boundaries but whose density is the function of x given below.  The plate is pictured at the right, colored to reflect the density function.  Here is a DPGraph picture of the plate colored based on the density function.

Maple Worksheet for the two center of mass examples

Find the total fluid force on a vertical circular porthole of radius 2 feet whose center is 14 feet below the surface of water whose weight is 62.4 pounds per cubic foot.

Find the total fluid force on one side of a vertical isosceles triangle whose base is 10 feet wide, height is 12 feet, and the base is resting on the bottom of a tank filled with water to a depth of 16 feet.  The weight of the water is 62.4 pounds per cubic foot.