EXAMPLE 5.16 Estimating Work by Counting Boxes GOAL Use the graphical method and Fp (N) F (N) 100 counting boxes to estimate the work 10.0 done by a force. 8.0 80 PROBLEM Suppose the force 6.0 60 applied to stretch a thick piece of 4.0 40 elastic changes with position as indicated in figure a. Estimate the 2.0 20 work done by the applied force. x (m) 0.2 0.4 0.6 0.8 1.0 x (m) 0.7 0.0 0.1 0.3 0.5 STRATEGY To find the work, simply (a) (b) count the number of boxes underneath the curve and multiply that number by the area of each box. The curve will pass through the middle of some boxes, in which case only an estimated fractional part should be counted. SOLUTION There are 62 complete or nearly complete boxes under the curve, 6 boxes that are about half under the curve, and a triangular area from x = 0 m to x = 0.10 m that is equivalent to 1 box, for a total of about 66 boxes. Because the area of each box is 0.10 J, the total work done is approximately 66 x 0.10 J = 6.6 J. LEARN MORE REMARKS Mathematically, there are a number of other methods for creating such estimates, all involving adding up regions approximating the area. To get a better estimate, make smaller boxes. QUESTION In developing such an estimate, is it necessary for all boxes to have the same length and width? (Select all that apply.) O No, if we have some idea what area the average box has and use the average area as an estimate. O No, because we are just counting boxes, not estimating area. O Yes, because we need square boxes to count accurately. O No, because counting boxes would work in exactly the same way if the boxes are rectangles but not squares. O Yes, because otherwise the number of boxes is not proportional to the area.

EXAMPLE 5.16 Estimating Work by Counting Boxes GOAL Use the graphical method and Fp (N) F (N) 100 counting boxes to estimate the work 10.0 done by a force. 8.0 80 PROBLEM Suppose the force 6.0 60 applied to stretch a thick piece of 4.0 40 elastic changes with position as indicated in figure a. Estimate the 2.0 20 work done by the applied force. x (m) 0.2 0.4 0.6 0.8 1.0 x (m) 0.7 0.0 0.1 0.3 0.5 STRATEGY To find the work, simply (a) (b) count the number of boxes underneath the curve and multiply that number by the area of each box. The curve will pass through the middle of some boxes, in which case only an estimated fractional part should be counted. SOLUTION There are 62 complete or nearly complete boxes under the curve, 6 boxes that are about half under the curve, and a triangular area from x = 0 m to x = 0.10 m that is equivalent to 1 box, for a total of about 66 boxes. Because the area of each box is 0.10 J, the total work done is approximately 66 x 0.10 J = 6.6 J. LEARN MORE REMARKS Mathematically, there are a number of other methods for creating such estimates, all involving adding up regions approximating the area. To get a better estimate, make smaller boxes. QUESTION In developing such an estimate, is it necessary for all boxes to have the same length and width? (Select all that apply.) O No, if we have some idea what area the average box has and use the average area as an estimate. O No, because we are just counting boxes, not estimating area. O Yes, because we need square boxes to count accurately. O No, because counting boxes would work in exactly the same way if the boxes are rectangles but not squares. O Yes, because otherwise the number of boxes is not proportional to the area.