CHAPTER 2 2.4 Continuity Applications to Physics and Engineering Work: W = lim n->   i=1 n f (x i * )  x =  a b f (x) dx Moments and centers of mass:

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Presentation transcript:

CHAPTER Continuity Applications to Physics and Engineering Work: W = lim n->   i=1 n f (x i * )  x =  a b f (x) dx Moments and centers of mass: 1. M y = lim n->   i=1 n  x i f (x i )  x =   a b x f (x) dx 2. M x = lim n->   i=1 n  (1/2) [ f (x i )] 2  x =   a b (1/2) ) [ f (x)] 2 dx x = (1/A)  a b x f (x) dx y = (1/A)  a b (1/2)[ f (x)] 2 dx

Example: When a particle is located at a distance x meters from the origin, a force of cos (  x /3) newtons act on it. How much work is done in moving the particle from x = 1 to x = 2? Interpret your answer by considering the work done from x = 1 to x = 1.5 and from x = 1.5 to x = 2. Example: Show how to approximate the required work by Riemann sum. Then express the work as an integral and evaluate it. A uniform cable hanging over the edge of a tall building is 40ft. long and weighs 60 lb. How much work is required to pull 10ft. of the cable to the top?

Example: A circular swimming pool has a diameter of 24ft, the sides are 5ft high, and the dept of the water is 4ft. How much work is required to pump all of the water out over the size? (use the fact that water weighs 62.5 lb/ft 3.) Example: x = The masses m i are located at the points P i. Find the moments M x and M y and the center of mass of the system. m 1 = 3, m 2 = 3, m 3 = 8, m 4 = 6; P 1 =(0,0), P 2 =(1,8), P 3 =(3,-4), P 4 =(-6,-5).