# Definite Integrals Questions and Answers

Calculus

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

Calculus

Definite IntegralsIf 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

Calculus

Definite IntegralsUse 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

Calculus

Definite Integrals1 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

Calculus

Definite IntegralsA 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

Calculus

Definite Integrals20 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

Calculus

Definite Integralspic 12 Quiz 1 0 Of g x dx converges Of g x dx diverges Of h x dx converges Of h x dx diverges

Calculus

Definite Integrals110 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

Calculus

Definite Integrals3 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

Calculus

Definite IntegralsProject 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

Calculus

Definite IntegralsWhich 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

Calculus

Definite Integralsy 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

Calculus

Definite Integrals1 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

Calculus

Definite IntegralsFigure 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

Calculus

Definite IntegralsThe 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

Calculus

Definite Integrals4 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

Calculus

Definite IntegralsThe 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

Calculus

Definite IntegralsFind 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

Calculus

Definite IntegralsSketch 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

Calculus

Definite IntegralsEvaluate 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

Calculus

Definite IntegralsY 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

Calculus

Definite IntegralsSketch the region enclosed by the given curves 1 y 16x y X 0 16 10 8 y 10 8 aff 2 Find the area of the region Need Help Read It 10 Watch It 15 8 X X y 10 8 10 6 y 10 2 10 8 15 X X

Calculus

Definite IntegralsFind the values of c such that the area of the region bounded by the parabolas y 4x c and y c 4x2 is 288 Enter your answers as a comma separated list C

Calculus

Definite IntegralsFind the area of the shaded region 20 15 10 5 y 9x x 2 y 2x 4 6 7 14 00 8 10 X

Calculus

Definite IntegralsMY NOTES 6 ASK YOUR TEACHER Sketch the region enclosed by the given curves y 14x1 y x 5 4 2 y 25 20 15 10 5 y 15 10 5 10 Need Help Read It 15 Find the area of the region PRACTICE ANOTHER 2 4 6 X 6 6 4 4 2 y 15 10 5 10 15 y 25 20 15 10 5 2 4 4 6 6 X X

Calculus

Definite IntegralsThe total accumulated costs C t and revenues R t in thousands of dollars respectively for a photocopying machine satisfy 1 C t 55 and 46 R t 5t5e t6 where t is the time in years Find the useful life of the machine to the nearest year What is the total profit accumulated during the useful life of the machine The useful life of the machine is year s Round to the nearest year as needed F www Using the useful life of the machine rounded to the nearest year the total profit accumulated during the useful life of the machine is Round to the nearest dollar as needed

Calculus

Definite IntegralsCalculate the indicated Riemann sum S4 for the function f x 19 3x Partition 0 12 into four subintervals of equal length and for each subinterval XK 1Xk let ck 2xk 1 xk 3 S4 Simplify your answer CIE

Calculus

Definite Integrals2 Find the volume generated by revolving the regions bounded by the given curve about the x axis y x 1 x 0 x 3 y 0

Calculus

Definite IntegralsClick here to watch the video Given vectors u and v find a 3u b 3u 5v c v 5u u 4 5 v 2 0 a 3u CND

Calculus

Definite Integrals5 The elevation of a path is given by f x x 5x 30 measured in feet where measures horizontal distances in miles Draw a graph of the elevation function and find its average value for 0 x 4

Calculus

Definite IntegralsA company determines that its marginal revenue per day is given by R t where R t is the total accumulated revenue in dollars on the fth day The company s marginal cost per day is given by C C where C t is the total accumulated cost in dollars on the fth day R t 90e R 0 0 C t 90 0 8t C O 0 a Find the total profit P T from t 0 to t 10 the first 10 days P T R T C T S R C 0 at dt The total profit is Round to the nearest cent as needed b Find the average daily profit for the first 10 days The average daily profit is Round to the nearest cent as needed

Calculus

Definite IntegralsFind the area between the curve and the x axis over the indicated interval y 121 x 11 11 The area under the curve is Simplify your answer

Calculus

Definite IntegralsFind the area of the region bounded by the graphs of the given equations y x y x 3333333 A B 1 2 X Mi x dx dx 0 25 0 25 G

Calculus

Definite IntegralsFind the area of the region bounded by the graphs of the given equations y 4x y x x 0 x 2 Set up the integral s that will give the area of the region Select the correct choice below and fill in any answer box es to complete the choice OA OB S Dax dx O C fo dx dx

Calculus

Definite IntegralsEvaluate the limit lim 3 tan 4x Enter a simplified multiple of or o or 818

Calculus

Definite Integralsdp The marginal price dx at x units of demand per week is proportional to the price p There is no weekly demand at a price of 150 per unit p 0 150 and there is a weekly demand of 7 units at the price of 119 96 per unit p 7 119 96 Use the given information to answer the questions a Find the price demand equation p x Type an exact answer in simplified form Round all decimal values to the nearest hundredth as needed

Calculus

Definite IntegralsThe marginal cost in dollars per square foot of installing x square feet of kitchen countertop is given by C x x a Find the cost of installing 50 ft2 of countertop b Find the cost of installing an extra 17 ft2 of countertop after 50 ft2 have already been installed a Set up the integral for the cost of installing 50 ft2 of countertop C 50 dx The cost of installing 50 ft of countertop is Round to the nearest cent as needed b Set up the integral for the cost of installing an extra 17 ft2 after 50 ft has already been installed C 50 17 C 50 dx 50 ELDB The cost of installing an extra 17 ft2 of countertop after 50 ft has already been installed is Round to the nearest cent as needed 4

Calculus

Definite IntegralsFind s t where s t represents the position function v t represents the velocity function and a t represents the acceleration function a t 6t 4 with v 0 7 and s 0 5 t SOCIED

Calculus

Definite Integrals4 12 pts Based on historical data for population growth rates and the initial number of eagles present in a region biologists expect an eagle population to grow according to the model 200 P t 1 82 33e 0 162t where t is measured in years a How many eagles were initially present b When will the population reach 100 eagles Give exact and approximate solutions

Calculus

Definite IntegralsSuppose that f is the function given in the graph below and that f is a piecewise function whose parts are either portions of lines or portions of circles as pictured 6 3 4 3 A 6 2 A 1 1 6 5 4 3 2 1 2 3 5 6 Let A be the function defined by A 2x f t dt b Compute the followings 5 q a What does the Second FTC tell us about the relationship between A and f Write an equation to describe the relationship 0 1 pt 30 29

Calculus

Definite Integralsroblem 3 Evaluate the following integral Problem 3 3 6 cot x In sin x dx Enter yo as in the

Calculus

Definite Integralsroblem 1 Evaluate the following integral 4 x 2 x 5 dx Don t forget the absolute value in your answer Problem 1 Enter your answer as a symbolic function of x as in these examples Don t forget the absolute in your answer Use abs for absolute v