Exam 5: Integration

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The maker of an automobile advertises that it takes 13 seconds to accelerate from 5 kilometers per hour to 65 kilometers per hour. Assuming constant acceleration, compute the acceleration in meters per second per second. Round your answer to three decimal places. ​

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Use left endpoints and 12 rectangles to find the approximation of the area of the region between the graph of the function Use left endpoints and 12 rectangles to find the approximation of the area of the region between the graph of the function   and the x-axis over the interval   . Round your answer to four decimal places. and the x-axis over the interval Use left endpoints and 12 rectangles to find the approximation of the area of the region between the graph of the function   and the x-axis over the interval   . Round your answer to four decimal places. . Round your answer to four decimal places.

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Find the time required for an object to cool from Find the time required for an object to cool from   F to   F by evaluating   where t is time in minutes. Round your answer to four decimal places. ​ F to Find the time required for an object to cool from   F to   F by evaluating   where t is time in minutes. Round your answer to four decimal places. ​ F by evaluating Find the time required for an object to cool from   F to   F by evaluating   where t is time in minutes. Round your answer to four decimal places. ​ where t is time in minutes. Round your answer to four decimal places. ​

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Evaluate the integral. ​ Evaluate the integral. ​   ​ Given, ​   ,   ,   ,   . ​ ​ Given, ​ Evaluate the integral. ​   ​ Given, ​   ,   ,   ,   . ​ , Evaluate the integral. ​   ​ Given, ​   ,   ,   ,   . ​ , Evaluate the integral. ​   ​ Given, ​   ,   ,   ,   . ​ , Evaluate the integral. ​   ​ Given, ​   ,   ,   ,   . ​ . ​

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Use the differential equation Use the differential equation   and the initial condition   to find y. and the initial condition Use the differential equation   and the initial condition   to find y. to find y.

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Find an equation for the function Find an equation for the function   whose derivative is   and whose graph passes through the point   . whose derivative is Find an equation for the function   whose derivative is   and whose graph passes through the point   . and whose graph passes through the point Find an equation for the function   whose derivative is   and whose graph passes through the point   . .

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Find the following indefinite integral. ​ Find the following indefinite integral. ​   ​ ​

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Find the indefinite integral. ​ Find the indefinite integral. ​   ​ ​

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The graph of f consists of line segments, as shown in the figure. Evaluate the definite integral The graph of f consists of line segments, as shown in the figure. Evaluate the definite integral   using geometric formulas.   ​ using geometric formulas. The graph of f consists of line segments, as shown in the figure. Evaluate the definite integral   using geometric formulas.   ​ ​

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The rate of disbursement The rate of disbursement   of a 2 million dollar federal grant is proportional to the square of   . Time t is measured in days   and Q is the amount that remains to be disbursed. Find the amount that remains to be disbursed after 40 days. Assume that all the money will be disbursed in 100 days. ​ of a 2 million dollar federal grant is proportional to the square of The rate of disbursement   of a 2 million dollar federal grant is proportional to the square of   . Time t is measured in days   and Q is the amount that remains to be disbursed. Find the amount that remains to be disbursed after 40 days. Assume that all the money will be disbursed in 100 days. ​ . Time t is measured in days The rate of disbursement   of a 2 million dollar federal grant is proportional to the square of   . Time t is measured in days   and Q is the amount that remains to be disbursed. Find the amount that remains to be disbursed after 40 days. Assume that all the money will be disbursed in 100 days. ​ and Q is the amount that remains to be disbursed. Find the amount that remains to be disbursed after 40 days. Assume that all the money will be disbursed in 100 days. ​

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Evaluate the Evaluate the   using L'Hôpital's Rule if necessary. ​ using L'Hôpital's Rule if necessary. ​

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Evaluate the definite integral. ​ Evaluate the definite integral. ​   ​ ​

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Find the indefinite integral. ​ Find the indefinite integral. ​   ​ ​

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Evaluate the limit Evaluate the limit   first by using techniques from Chapter 3 then by using L'Hopital's Rule. ​ first by using techniques from Chapter 3 then by using L'Hopital's Rule. ​

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Determine the area of the given region. ​ Determine the area of the given region. ​   ​   ​ ​ Determine the area of the given region. ​   ​   ​ ​

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Write the following limit as a definite integral on the interval [2, 8] where Write the following limit as a definite integral on the interval [2, 8] where   is any point in the   subinterval. ​   ​ is any point in the Write the following limit as a definite integral on the interval [2, 8] where   is any point in the   subinterval. ​   ​ subinterval. ​ Write the following limit as a definite integral on the interval [2, 8] where   is any point in the   subinterval. ​   ​ ​

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Find the indefinite integral of the following function. ​ Find the indefinite integral of the following function. ​   ​ ​

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Find the indefinite integral. ​ Find the indefinite integral. ​   ​ ​

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Find the indefinite integral of the following function. ​ Find the indefinite integral of the following function. ​   ​ ​

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Evaluate the limit Evaluate the limit   using L'Hopital's Rule if necessary. ​ using L'Hopital's Rule if necessary. ​

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