Applications of Integration

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Presentation transcript:

Applications of Integration Section 7.5a

Work Done Lifting A leaky bucket weighs 22 N empty. It is lifted from the ground at a constant rate to a point 20m above the ground by a rope weighing 0.4 N/m. The bucket starts with 70 N (approx. 7.1 L) of water, but it leaks at a constant rate and just finishes draining as the bucket reaches the top. Find the amount of work done (a) lifting the bucket alone; (b) lifting the water alone; (c) lifting the rope alone; (d) lifting the bucket, water, and rope together. Remember that work is the area under the force graph…

Work Done Lifting (a) lifting the bucket alone The bucket’s weight is constant, so we need to exert the 22 N force along the entire 20m distance: F 22N Work 20m x

Work Done Lifting (b) lifting the water alone The force needed to lift the water is equal to its weight, which decreases steadily from 70 N to 0 N over the 20m lift: F 70N Work 20m x

Work Done Lifting (c) lifting the rope alone The force to lift the rope also varies, starting at 8 N and ending at 0 N: F 8N Work 20m x

Work Done Lifting (d) lifting the bucket, water, and rope together Simply sum for the total work done:

Work Done Pumping The conical tank shown is filled to within 2 ft of the top with olive oil weighing 57 . How much work does it take to pump the oil to the rim of the tank? y Volume of one of these thin slabs: x

Work Done Pumping The conical tank shown is filled to within 2 ft of the top with olive oil weighing 57 . How much work does it take to pump the oil to the rim of the tank? The force to lift this slab is equal to its weight: This force must be applied over a distance of To calculate the work of moving all of the slabs, use a Riemann sum from y = 0 to y = 8…

Work Done Pumping The conical tank shown is filled to within 2 ft of the top with olive oil weighing 57 . How much work does it take to pump the oil to the rim of the tank?

Hooke’s Law Revisited A bathroom scale is compressed 1/16 in. when a 150-lb person stands on it. Assuming the scale behaves like a spring that obeys Hooke’s Law. How much does someone who compresses the scale 1/8 in. weigh? with ,

Hooke’s Law Revisited A bathroom scale is compressed 1/16 in. when a 150-lb person stands on it. Assuming the scale behaves like a spring that obeys Hooke’s Law. (b) How much work is done in compressing the scale 1/8 in.?