Exam 5: Integration

The sales S (in thousands of units) of a seasonal product are given by the model where t is the time in months, with corresponding to January. Find the average sales for the first quarter . Round your answer to three decimal places. ​

Find the indefinite integral . ​

Evaluate the integral . ​

The diagram below shows upper and lower sums for the function ​ ​ Using 4 subintervals. Use upper and lower sums to approximate the area of the region using the 4 subintervals. ​ ​

Find the sum given below. ​ ​

Find given ​ . ​

Find the derivative of the function . ​

Solve the differential equation. ​ ​

The rate of depreciation of a machine is inversely proportional to the square of where V is the value of the machine t years after it was purchased. The initial value of the machine was $500,000 and its value decreased $100,000 in the first year. Estimate its value after 4 years. Round your answer to the nearest integer. ​

Find the indefinite integral . ​

Evaluate the following definite integral by the limit definition. ​ ​

Evaluate the definite integral of the algebraic function. ​ ​ Use a graphing utility to verify your results. ​

Sketch the region whose area is given by the definite integral and then use a geometric formula to evaluate the integral. ​ ​

Evaluate the definite integral. ​ ​

Find the sum given below. ​ ​

Find the average value of the function over the interval .

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

A ball is thrown vertically upwards from a height of 8 ft with an initial velocity of 30 ft per second. How high will the ball go? Note that the acceleration of the ball is given by feet per second per second.

Find the indefinite integral . ​

Find the indefinite integral. ​ ​