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# P27 - f x = √ 100-x 2 on-10 10 but what geometric ﬁgure...

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Project 27 MAC 2311 1. Examine the function f ( x ) = 4 x 2 - x + 2 x on ( 0 , ) . Find any one antiderivative for f ( x ) . What is the general antiderivative for f ( x ) ? Calculate the indefinite integral Z f ( x ) d x . Calculate the exact area under f ( x ) on [ 1 , 2 ] . Note: justify that the function is always positive on the interval using quadratic formula. Suppose the rate (in liters per hour) at which water runs into a trough during a rainstorm is given by f ( t ) , where t represents the time in hours after the rainstorm begins. What is the exact change in the volume of water in the trough as the storm continues from four to eight hours in duration? Combine any logarithms contained in your answer into a single logarithm. If the trough contains 20 liters at the after 1 hour of the storm, find the function V ( t ) that represents the amount of water in the tank after t hours.

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2. Sometimes it is difficult (or impossible) to find a useful antiderivative in order to evaluate a definite integral. You can still use approximations or areas when this happens. For example, we have not learned a useful antiderivative for the function
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Unformatted text preview: f ( x ) = √ 100-x 2 on [-10 , 10 ] but what geometric ﬁgure does it represent? If a population of geese is changing at a rate of √ 100-t 2 geese per year, where t is the time (in years) from the present, then what will be the growth in population over the next ten years? (Use the geometry to evaluate the integral.) 3. A machine part oscillates back and forth in a straight line with acceleration function given by ‘ ( t ) = cos(2 t )-sin( t ) m/s 2 , where t is the time elapsed in seconds. If its initial velocity is 1 m/s, write a formula for its velocity at time t . On what subinterval(s) of [ 0 , 2 π ] is the velocity positive? negative? (Use the trigonometric identity sin(2 t ) = 2 sin( t ) cos( t ) and remember that-1 ≤ sin( t ) ≤ 1 .) Find both the displacement and total distance traveled by the part on the time interval [ 0 , 2 π ] by applying Net Change Theorem....
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