PracticeFinal.pdf - MAT1322 Final exam Practice sheet...

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MAT1322, Final exam Practice sheet:IntegrationArea, Volumes, Applications of IntegralsCompute the area of the following regions:1. Bounded region delimited byy=ex, y= 1, x= 2.2. Bounded region delimited byy= sin(x), y= cos(x), x= 0, y= 0, x=fi2.3. Bounded region delimited byy= 3x6, y= 9/x, x= 4.Compute the volume of the following solids:1.y=ex, y= 1, x= 2 rotated aroundy=2.2.y= sin(x), y= cos(x), x= 0, y= 0, x=Ô22abouty=1.3.y= 3x6, y= 9/x, y= 0, x= 4 aboutx=2.Applications of integrals:1. A cable that weighs 2kg/mis used to lift 800kgof coal up to a mine shaft which is 500mdeep. Find the work done.2. Suppose that 2 J of work is needed to stretch a spring from its natural length of 30 cm to alength of 42 cm. How much work is needed to stretch the spring from 35 cm to 40cm? Hint:you may use Hooke’s law, F=kx to computekfrom the numbers given and then apply thisto the question.3. A leaky 10 kg bucket is lifted from the ground to a height of 12 m at a constant speed with arope that weighs 0.8 kg/m. Initially the bucket contains 36 kg of water, but the water leaksat a constant rate and finishes draining just as the bucket reaches the 12 m level. How muchwork is done?4. A tank with a capacity of 500 liters contains 500 liters of water with 100 kg of salt in solution.Water containing 1 kg of salt per liter is entering at the rate of 3 liters per minute, and themixture is allowed to flow out of the tank at a rate of 3 liters per minute. Find the amountof salt in the tank after 5 minutes.5. A tank in the shape of an inverted cone has a height of 15 m and a base radius of 4 m and isfilled with water to a depth of 12 meters. Determine the amount of work needed to pump allof the water to the top of the tank. Assume that the density of the water is 1000

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