1. Section 6.4 Another Application of Integration

2. Definition: Work Work generally refers to the amount of effort required to perform a task

3. More precisely… If an object is moved a distance d in the direction of an applied force F, the work done by the force is W=Fd Examples? A force pushing or pulling an object The downward pull of gravity on an object

4. More details! If the object moves along a straight line with position s(t) then the force F acting on the object in the same direction is defined by Newton’s second law: F = (mass)(acceleration) = ms’’(t)

5. Remarks: F=ms’’(t) Mass has units in kilograms Distance has units in meters Time has units in seconds F has units in (kg)(m)/s 2 = N (Newton) –In the US, Force may use units of weight (pounds) W=Fd gives units of Newton-meters or Joules

6. Example How much work is done in lifting a 1.2 kg book off the floor to put it on a desk that is 7 m high? (assume g = 9.8 m/s 2 )

7. What if the force is not constant? Suppose an object moves along a straight line from x = a to x = b by a varying force f(x). Partition [a,b] into subintervals of length Choose a sample point Since f(x) is a varying force and we’ll assume that is “small,” we can say that f(x) is almost constant over So the force acting on the object over is approximately

8. Work! So the work done to move the particle from And so the total work is

9. Reimann Sum!

10. Example When a particle is a distance x from the origin, a force of pounds acts on it. How much work is done to move the object from x=1 to x=3?

11. A more exciting example: Work required to move a liquid Suppose a tank is shaped like an inverted circular cone with a radius of 4 meters at the top and a height of 10 meters. The tank is filled to a height of 8 meters. Find the work required to empty the tank by pumping the water out the top. Use the fact that the density of water is 1000 kg/m 3

12. A drawing almost always helps 4 8 10

13. Another Example A spherical tank with a radius of 8 ft is half full of a liquid that weighs 50 pounds/ft 3. Find the work required to pump the liquid out of a hole in the top of the tank.