v ( t ) = 5t 8, 0 t 3

Transcription

1 Use the Fundamental Theorem of Calculus to evaluate the integral. 27 d 8 2 Use the Fundamental Theorem of Calculus to evaluate the integral. 6 cos d 6 The area of the region that lies to the right of the y ais and to the left of the parabola = 6y y 2 (the shaded region in the figure) is given by the integral 6 ( 6y y 2 )dy. (Turn your head clockwise and think of the region as lying below the curve = 6y y 2 from y = to y = 6.) Find the area of the region. Epress your answer as a fraction, where necessary. Evaluate the integral. 5 4 sec tan d 4 Evaluate the integral. e d 5 Find the eact area of the region that lies beneath the given curve. y = 5sin( ), 6 7 The velocity function (in meters per second) is given for a particle moving along a line. v ( t ) = 5t 8, t (a) Find the displacement (in meters) of the particle during the given time interval. Epress your answer as a fraction, where necessary. (b) Find the distance (in meters) traveled by the particle during the given time interval. Epress your answer as a fraction, where necessary. 8 The linear density of a rod of length 64 m is given by ( ) = measured in kilograms per meter, where is measured in meters from one end of the ro Find the total mass of the ro Water flows from the bottom of a storage tank at a rate of r (t ) = 24 6t liters per minute, where t 4. Find the amount of water that flows from the tank during the first 6 minutes. liters PAGE

2 The velocity of a car was read from its speedometer at 5 second intervals and recorded in the table. Use the Midpoint Rule to estimate the distance traveled by the car. Use 5 equal subintervals. t (s) v (mi / h) Which of the following shaded regions represents g ( ) = ( 6 + t ) dt? Please round the answer to one decimal place. miles..to be continued PAGE 2

3 continuation 2 Use Part of the Fundamental Theorem of Calculus to find the derivative of the function. Place parenthesis around the angle. y = e cos t dt At what values of does F ( ) = 2 sin t t dt have local maimum values? e. [The integrand f (t ) = 2 sin t is not defined when t t =, but we know that its limit is 2 when t. So we define f () = 2 and this makes f a continuous function everywhere. ] =, 2,, 4, 5, 6,... =,, 5, 7,,... = 2, 4, 6, 8,... Find g' ( ) by using the fact that if f is continuous on [a, b ] and g ( ) = f (t )dt, then g' ( ) = f ( ). a =,, 2,, 4, 5,... e. =, 2,, 4, 5, 6,... Find g' ( ) by using the fact that if f is continuous b on [a, b ] then f ( )d = F (b ) F (a ), a where F is any antiderivative of f, that is, F' = f. PAGE

4 4 A high tech company purchases a new computing system whose initial value is V. The system will depreciate at the rate f = f (t ) and will accumulate maintenance costs at the rate g = g (t ), where t is the time measured in months. The company wants to determine the optimal time to replace the system. Suppose that Evaluate the indefinite integral. Be sure to enclose all angles in parentheses. sec tan d 2 Evaluate the integral. f (t ) = { V 25 V 25 t if < t 5 if t > 5 e d e + 8 and g (t ) = Vt 2 24 t > Determine the length of time T depreciation D (t ) = value V. t f (s )ds T = months for the total to equal the initial 5 Evaluate the integral by making the given substitution. 2 Evaluate the definite integral. Your answer should be a numerical value csc t cot t dt 22 Evaluate the definite integral. Simplify your answer. m m 2 2 d 5sin d, u = 2 Evaluate the integral. 6 Evaluate the integral by making the given substitution. e 8 e 4 d ln e cos sin d, u = cos 7 Evaluate the indefinite integral. d ( 2 + 8) 8 8 Evaluate the indefinite integral. sin t dt PAGE 4

5 24 Alabama Instruments Company has set up a production line to manufacture a new calculator. The rate of production of these calculators after t weeks is d = dt 6 (t + ) 2 calculators/week (Notice that production approaches 6 per week as time goes on, but the initial production is lower because of the workers' unfamiliarity with the new techniques.) Find the number of calculators produced from the beginning of the fourth week to the end of the fourth week. 27 Evaluate the integral. p 5 ln p dp 5 p 6 ln p 25 p 6 + C 6 p 6 ln p 6 p 6 e. + C 6 p 6 ln p 6 p 6 + C 6 p 6 ln p + 25 p 5 + C 25 If f is continuous and f ( )d = 2, find 2 f ( 2 )d. 26 Evaluate the integral p 6 ln p + 5 p 6 + C 28 Evaluate the integral. t ln t dt e 5 d 52 ln ln ln ln e. 24 ln PAGE 5

6 2 First make a substitution and then use integration by parts to evaluate the integral. e 8 d 8e + 2e 6e 6e + 2 A rocket accelerates by burning its onboard fuel, so its mass decreases with time. Suppose the initial mass of the rocket at liftoff (including its fuel) is m, the fuel is consumed at rate r, and the ehaust gases are ejected with constant velocity v (relative to the rocket). A e model for the velocity of the rocket at time t is given by the equation v ( t ) = gt v ln m rt e m 8e 6e e. 2e + e 7e e where g is the acceleration due to gravity and t is not too large. If g =.8 m/s 2, m = 27 kg, r = 5 kg/s, and v = 2 m/s, find the height of e the rocket two minutes after liftoff. Please round the answer to the nearest integer. m Evaluate the indefinite integral. 8 ln d Use the reduction formula below to answer the questions that follow. sin n d = n sin n cos + n n sin n 2 d (a) Evaluate 7sin 2 d. (b) Use part (a) to evaluate 7sin 4 d. PAGE 6