Practice Using Definite Integrals 1. Power companies typically bill customers based on the number of kilowatt-hours used during a single billing period. A kilowatt is a measure of how much power (energy) a customer is using, while a kilowatt-hour is one kilowatt of power being used for one hour. For constant power use, the number of kilowatt-hours used is calculated by kilowatt-hours = kilowatts . time (in hours). Thus, if customers use 5 kilowatts for 30 minutes, they'll have used 5 kilowatts . - hours = 2.5 kilowatt-hours. Suppose the power use of a customer over a 30-day period is given by the continuous function P = (), where P is kilowatts, : is time in hours, and / = 0 corresponds to the beginning of the 30 day period. A. Approximate, with a Riemann sum, the total number of kilowatt-hours used by the customer in the 30-day period, and explain why your Riemann sum is an approximation of the desired property. B. Derive an expression representing the total number of kilowatt-hours used by the customer in the 30-day period, and explain your reasoning. (This expression should not be an approximation.) C. Consider the following table of data for the function f(().Practice Using Definite Integrals 0 2.3 2.5 2 2.1 3 3.9 5 5.5 6 4.5 7 5.6 1.2 9 1.0 10 1.8 Recall that () represents the number of kilowatts being used by a customer at time : hours from the beginning of the billing period. Estimate the number of kilowatt-hours the customer uses in this 10-hour period, and explain your method. 2. Two separate rabbit populations are observed for 80 weeks, starting at the same time and with the same initial populations. The growth rates of two rabbit populations are modeled as follows, where / = ( corresponds to the beginning of the observation period: ri(t) = 4sin( #7/) +0. 1/+ 1, where r, is rabbits per week, and : is time in weeks, my(1) = 1/2, where r2 is rabbits per week, and / is time in weeks. Below is a graph of the curves representing the rates of growth of the two populations over the observation period:Practice Using Definite Integrals 10 20 30 40 50 60 70 80 A. Using your calculator, find (approximately) the first positive time / for which the rates of growth for the two populations are the same.B. What's the physical significance of the area between the two curves from time / = ( until the first time where the two rates are the same? What does the area represent? C. Suppose you want to find the first time (call it ?) after the beginning of the observation period at which the two rabbit populations have identical populations. Write an equation to solve for the unknown variable T. D. Simplify your equation from part C until you can use your calculator on it. Then use your calculator to solve this equation for 7. 3. A bucket that weighs 3 lb and a rope of negligible weight are used to draw water from a well that's 60 feet deep. Suppose the bucket starts with 37 lb of water and is pulled up by a rope at 2 ft/sec, while water leaks out of the bucket at a rate of -Ib/sec. A. How long does it take for the bucket to get to the top of the well? Write an equation that expresses the total weight of the bucket as a function of time, as the time varies from 0 until the time the buckets gets to the top.B. Recall that work equals force times distance. Calculate the work done in lifting the bucket to the top of the well, keeping in mind that here force is equal to weight. 4. A certain computer algorithm used to solve very complicated differential equations uses an iterative method. That is, the algorithm solves the problem the first time very approximately, and then uses that first solution to help it solve the problem a second time just a little bit better, and then uses that second solution to help it solve the problem a third time just a little bit better, and so on. Unfortunately, each iteration (each new problem solved by using the previous solution) takes a progressively longer amount of time. In fact, the amount of time it takes to process the k-th iteration is given by 7(k) = 1.2* + 1 seconds. A. Use a definite integral to approximate the time (in hours) it will take the computer algorithm to run through 60 iterations. (Note that 7() is the amount of time it takes to process just the *-th iteration.) Explain your reasoning. B. The maximum error in the computer's solution after & iterations is given by Error = 2k-2. Approximately how long (in hours) will it take the computer to process enough iterations to reduce the maximum error to below 0. 0001