Definite Integrals Questions and Answers

Find the area between y 9 and y x 1 with 0 a

Recall that cot x cos x sin x

Acompany is considering a new manufacturing process It knows that the rate of savings in dollars per year from the process will be about S t 4000 t 4 where t is the number of years the process has been in use Find the total savings during the first year Find the total savings during the first 8 years The total savings during the first year is Simplify your answer The total savings during the first 8 years is Simplify your answer

Use the given transformation to evaluate the integral f S x y dA where R is the square with vertices 0 0 4 6 6 4 10 2 and x 4u 6v y 6u 4v

6 Estimate the value of 1 n 0 to within 5 decimal places n x n 1 2n 1 when x 0 5 Your estimate should be accurate

Find the present value of a single future payment of FV 35 000 to be made 8 years from today assuming an interest rate of 1 6 per year compounded continuously Identify the correct present value formula for the given values OA PV 35 000 e 0 016 8 OB 35 000 PV e 0 016 8 OC PV 35 000 e 0 016tdt 0 O D PV 35 000 e 1 6 8 4 PV Do not round until the final answer Then round to the nearest hundredth as needed

It takes a force of 43 764 lb to compress a coll spring assembly from its free height of 11 inches to its fully compressed height of 5 inch a What is the assembly s force constant b How much work does it take to compress the assembly the first half inch the second half inch Answer to the rearest In lb a k lb in b How much work does it take to compress the assembly the first half inch in lb Round to the nearest whole number as needed How much work does it take to compress the assembly the second half inch in b Round in the nearest whole number as needed 301

Evaluate ff fr x y dV where T is the solid bounded by the cylinder x y 1 and the planes z 4 and z 8

Set up the following triple integral but do not calculate it a solid that lies within the cylinder x y 4 and above the xy plane and below the cone z 4x 4y Use cylindrical coordinates

Find s S cos 9x dx cos 9x dx sin 9x 9 Hint Did you include the constant of integration X

3 2 1 1 Y 2 1 0 2 graph of f x 3 4 5 6 7 8 9 f 0 f x has a local a Find f x dx dr b Find c Find d Use the graph of f x shown above to find the critical points of f x and to classify each as the location of a local maximum or local minimum f r has a local at the critical point z at the critical point x f x dx F x dx e Suppose f 3 5 Use the Fundamental Theorem of Calculus and the graph of f x shown above to find each of the following f 6

6 5 points The image below depicts which of the following approximations of 10 f x dx A A right hand sum with n 2 and Ar 6 B A left hand sum with n 6 and Ar 2 C A right hand sum with n 6 and Az 2 D A left hand sum with n 2 and Ax 6

4 5 points Which of the following is a true statement ab Y 6 4 2 f t LA 1 2 3 6 5 4 3 2 2 5 A Ax 8 5 B Ax 4 C Ax 8 D Ax 34 E Ax 32 12 X A on the interval x 20 the function f x has a global minimum at x 3 and a global maximum at x 2 B on the interval x 20 the function f x has a global maximum at x 2 and a global minimum at x 0 C on the interval x 20 the function f x has no global minimum and no global maximum D on the interval z 20 the function f x has a global maximum at x 2 and no global minimum 5 5 points What is Ar for either a left or right hand sum with n 4 rectangles for 6 5 points The image below depicts which of the following approximations of 34 24 f x dx 12 1 f x da A A right hand sum with n 2 and Ax 6 B A left hand sum with n 6 and Ax 2 C A right hand sum with n 6 and Ax D A left hand sum with n 2 and Ax 6

13 5 points The rate r t upon the amount of time t in hours that has elapsed since they r t dt 0 035 JO What is the percent blood alcohol concentration of the adult 3 hours after finishing drinking if their blood alcohol concentration was 0 079 percent at the time they finished their last drink Include units in your answer Answer 14 5 points Consider the functions f x x x 7 and g x 8x 7 a Find the intersection points of these two functions by setting f x g x and solving for x Answer b Express the area trapped between these two functions as a definite integral and then use a calculator to evaluate this definite integral Give your answer to four decimal places

Find the volume V of the described solid S The base of S is the region enclosed by the parabola y 2 3x2 and the x axis Cross sections perpendicular to the y axis are squares V 8 X

9 points Find the dimensions of a can i e a cylinder of radius r height and volume 1 m with minimal surface area Hint the volume V nr h an the surface area S 2 rh 2 r

1 Use integration by parts to find the integral In x ln x dx

E GENERAL PERMANENT ENDOWMENTS Find the size of the permanent endowment needed to generate an T yeart at continues rater annual 12 000 forever at a continuous interest rate of 6 Present value TC t e rt dt C Continous stream years

Solve the following equation for x log11 x 5 4 8 O a x 126 b x 116 Oc x 2043 d x 24 2

45t 6 A particle has a velocity modeled by v t 5t3 2t 4 What is the total distance traveled from t 1 tot 2 1 Provide your answer below measured in cm s over the interval 1 st 2 You may either enter an exact answer or round your answer to the nearest tenth of a centimeter

Question A particle has a velocity modeled by v t 3 2ln t 5 measured in cm s over the interval 4 t 7 What is the total distance traveled from t 4 to t 7 You may either enter an exact answer or round your answer to the nearest tenth of a centimeter Provide your answer below

7 Find the area under the parabola y x 1 from x 2 to x 2

13 If speedometer readings for a motorcycle at 20 second intervals are given in the table below then the Riemann sum t sec v t ft sec a true b false 20 40 60 17 17 20 0 23 19 23 20 19 20 20 20 is necessarily an upper estimate for the distance traveled by the cyclist in one minute 23r 1

11 Find the area of the region enclosed by y 12 x and y x 6 a 36 b 70 c 18 d 72 e none of the above

a b 14 c 3 8 A particle moves along a line so that its velocity at time t is v t t 1 measured in meters per second Find the displacement of the particle during the time period 0 t 2 a 1 m b 0 m c 1 m d 2 m e 3 m

Evaluate the definite integral below Enter an exact answer Provide your answer below S 1 3x 4x 4 dx L 3x 3 4x 4 dx

If you were to use a substitution of u sin x to evaluate the definite integral below what would the new limits of integration be Enter an exact answer Provide your answer below The new lower bound is 1 6 sin x cos x dx and the new upper bound is

Use substitution to evaluate the definite integral given below Enter an exact answer Provide your answer below M 2 x sin 3x cos 3x dx x sin 3x cos 3x dx

1 06011 5 5 8 3 The answer above is NOT correct Find the exact length of the curve given by x 2t y 1 3 0 t 1 incorr

A mining company owns two mines These mines produce an ore that can be graded into two classes regular grade and low grade The company must produce at least 420 tons of regular grade and 540 tons of low grade ore per week The first mine produces 6 tons of regular grade and 18 tons of low grade ore per hour The second mine produces 18 tons of regular grade and 9 tons of low grade ore per hour The operating cost of the first mine is 20 000 per hour and the operating cost of the second mine is 9500 per hour The first mine can be operated no more than 37 hours a week and the second mine can be operated no more than 28 hours a week How many hours per week should each mine be operated to minimize the cost Note that the ALEKS graphing calculator can be used to make computations easier hour s Second mine hour s First mine

20 3x 13x13x 16x 5 x 5 1 A 3 2x 3x 1 B 3x 2r 3x 1 x 5 C 3x 2x 3x 1 10 x 5 4 x 5 D 3x 2x 3x 1 10 x 5

If cos x sin4 x A B cos 2x C cos 4x D cos 2x cos 4x then A B C D D G G

pic 12 Quiz 1 0 Of g x dx converges Of g x dx diverges Of h x dx converges Of h x dx diverges

Find f f x 3 5x

110 Find the collision probability of 1200 bytes IEEE 802 11 FH packet at 2 Mbps and the Bluetooth Assume dwell period of an 802 11 transmission to be 3 ms and Bluetooth packet duration 0 625 ms What is PER with IEEE 802 11 DS

3 25 Points Consider the region R bounded by y rr 25 below A 09876132 5 4 1 1 the z axi 2 2 a Setup the definite integral including bounds of integration to find the volume found by rotating R around the x axis and use your calculator to compute that value rounding to two decimal places b Setup the definite integral including bounds of integration to find the volume found by rotating R around the y axis and use your calculator to compute the value rounding to two decimal Jogos

4 10 Points Find the anti derivative of 1 1 x 13 dx

Express the equation in exponential form a log4 94 4 1 b log4 16 2

Project motion of a model rocket When a model rocket is launched the fuel burns for a few seconds accelerating the rocket upward After burnout the rocket coasts upward for awhile and then begins to fall A parachute pops out shortly after the rocket starts down in order to slow the rocket Use the graph provided to answer the questions below Velocity m s 100 80 60 40 20 0 20 40 Time s 10 12 A Was the rocket going up or down 5 seconds after it was launched How do you B When did the rocket reach its highest point C Estimate the maximum altitude D Estimate the average velocity over the first 6 seconds E Compute exactly and give a practical interpretation of Tv t dt F Compute exactly and give a practical interpretation of v 8 G Find the average acceleration over the first 6 seconds

Which of the following statements is TRUE by the Comparison Test Hint When x e we have In x 1 We know 1 cos x 1 O fro e10 X foo dx e10 x2 ln x COS X X In x e10 X da converges because of a X 150 1 1 21 dx converges X COS X converges because ef 00 de diverges because dx converges because dx e 0 x2 1 00 1 e10 x2 de converges because S 1 X converges 1 x dx diverges dx diverges dx converges

y tan x 4 1 Suppose you want to transform the graph of the function t y tan tan x 1 1 Which transformations should you perform into the graph of the function X O Reflect the graph of the first function across the x axis translate it units to the left and translate it 2 units up ST Reflect the graph of the first function across the x axis translate it units to the right and translate it 2 units up 4 units to the left and translate it 2 units up Reflect the graph of the first function across the line y 1 translate it Reflect the graph of the first function across the line y 1 translate it units to the right and translate it 2 units up

1 Given a function of g x on 2 6 and g x 0 on 2 1 g x dx 10 g x 0 on 1 4 g x dx 4 g x 0 on 4 6 g x dx 6 estimate the following integrals 3 points each D 2 lg x dx A L2g x dx B 6 g x dx c S g x g x dx E f lg x dx

Figure 1 Rectangular garden fencing rdener has a large open space and wants to use fencing to create a rectangular garden with 2 er gardens of equal size as shown in Figure 4 He has 1000 feet of fencing Write an equation A w ibing the total area of the garden in terms of the width w

The marriage rate in a certain country is approximated by M t 8 3e 0 022t where M t is the number of marriages per 1000 people t years after 200 4 a Find the total number of marriages per 1000 people in the country from 2000 to 2004 Note that this is given by M t dt 0 b Find the total number of marriages per 1000 people in the country between 2004 and 2016

4 Find the radius of gyration with respect to its axis of the solid generated by revolving the first quadrant region under the curve y 4 x about the y axis

The graph of f is shown below Evaluate each integral by interpreting it in terms of areas 1 f x dx 1 5 2 f 2 dun 0 3 f x dx Note You can click on the graph to enlarge the image

Find the number b such that the line y b divides the region bounded by the graphs of y 36x and y 9 into two regions wi equal area Round your answer to two decimal places b

Sketch the region enclosed by the given curves Decide whether to integrate with respect to x or y Draw a typical approximating rectangle 2 4 x 4 y x y 4 y 4 4 y 2 2 Find the area of the region 2 X X 2 1 1 1 2 2 2 X X

Evaluate the integral and interpret it as the area of a region 2 1 14 sin x 4 cos 2x dx Sketch the region 1 KIN NTH T KI4 10 2 X a 2 416 412 kter X X

Y NOTES y Sketch the region enclosed by the given curves 2 y G 4 2 4 0 6 ASK YOUR TEACHER y 5 cos 6x y 5 sin 12x x 0 0 05 PRACTICE ANOTHER Find its area 0 10 0 15 0 20 0 05 0 10 0 15 0 20 0 25 0 25 X 0 30 X 12 y st 4 3 2 1 y 6 0 05 0 10 0 15 0 20 0 25 5 4 In 3 X 2 1 0 05 0 10 0 15 X 0 20 0 25 0 30 X