Definite Integrals Questions and Answers
![1 Find the area between y 3x and y x from x 0 to x 2 The area is](https://media.kunduz.com/media/sug-question-candidate/20230626234531223370-5686493.jpg?w=256)
![Find the area between y x and y x from x 0 to x 0 Simplify your answer The area is](https://media.kunduz.com/media/sug-question-candidate/20230626233820188386-5686493.jpg?w=256)
Calculus
Definite IntegralsFind the area between y x and y x from x 0 to x 0 Simplify your answer The area is
![Find F as a function of x and evaluate it at x 3 x 7 and x 9 2 1 1 1 31 dt F x F x 2014 1 9 F 3 0 F 7 0 090703 X F 9 0 09876543](https://media.kunduz.com/media/sug-question-candidate/20230626232511535226-3917327.jpg?w=256)
Calculus
Definite IntegralsFind F as a function of x and evaluate it at x 3 x 7 and x 9 2 1 1 1 31 dt F x F x 2014 1 9 F 3 0 F 7 0 090703 X F 9 0 09876543
![A tank is full of water Find the work in 3 required to pump the water out of the spout Use 9 8 m s for g Use 1 000 kg m as the density of water Round your answer to the nearest whole number 35 91x10 3 m 9 m](https://media.kunduz.com/media/sug-question-candidate/20230626212629316578-4426144.jpg?w=256)
Calculus
Definite IntegralsA tank is full of water Find the work in 3 required to pump the water out of the spout Use 9 8 m s for g Use 1 000 kg m as the density of water Round your answer to the nearest whole number 35 91x10 3 m 9 m
![A 600 lb cable is 100 ft long and hangs vertically from the top of a tall building a How much work is required to lift the cable to the top of the building b How much work is required to pull up only 30 feet of the cable Solution W a Here we don t have a formula for the force function but we can use an argument similar to the one that led to the definition of work Let s place the origin at the top of the building and the x axis pointing downward as in the following figure 0 x7 100 We divide the cable into small parts with length Ax If x is a point in the ith such interval then all points in the interval are lifted by approximately the same amount namely x The cable weighs 6Ax Thus the work done on the ith part in foot pounds is 6x Ax 6Ax x distance force We get the total work done by adding all these approximations and letting the number of parts become large so Ax 0 W lim 515 Ar ax 7 1 6x Ax pounds per foot](https://media.kunduz.com/media/sug-question-candidate/20230626212718854128-4426144.jpg?w=256)
Calculus
Definite IntegralsA 600 lb cable is 100 ft long and hangs vertically from the top of a tall building a How much work is required to lift the cable to the top of the building b How much work is required to pull up only 30 feet of the cable Solution W a Here we don t have a formula for the force function but we can use an argument similar to the one that led to the definition of work Let s place the origin at the top of the building and the x axis pointing downward as in the following figure 0 x7 100 We divide the cable into small parts with length Ax If x is a point in the ith such interval then all points in the interval are lifted by approximately the same amount namely x The cable weighs 6Ax Thus the work done on the ith part in foot pounds is 6x Ax 6Ax x distance force We get the total work done by adding all these approximations and letting the number of parts become large so Ax 0 W lim 515 Ar ax 7 1 6x Ax pounds per foot
![2 The graph off is shown in the figure The shaded region A has an area of 7 5 and ff x dx 16 5 Use this information to answer the following A 4 B 11 2 3 4 5 6 a f f x dx b ff x dx c Solf x dx d f3 f x dx](https://media.kunduz.com/media/sug-question-candidate/20230626200840701529-3917327.jpg?w=256)
Calculus
Definite Integrals2 The graph off is shown in the figure The shaded region A has an area of 7 5 and ff x dx 16 5 Use this information to answer the following A 4 B 11 2 3 4 5 6 a f f x dx b ff x dx c Solf x dx d f3 f x dx
![3 Find the area of the region bounded by the graphs of the equations y 1 x y 0 x 0 and x 8](https://media.kunduz.com/media/sug-question-candidate/20230626200900303672-3917327.jpg?w=256)
Calculus
Definite Integrals3 Find the area of the region bounded by the graphs of the equations y 1 x y 0 x 0 and x 8
![If the function y e x is vertically compressed by a factor of 4 reflected across the x axis and then shifted up 5 units what is the resulting function Write your answer in the form y ceax b Provide your answer below](https://media.kunduz.com/media/sug-question-candidate/20230626164033767519-5066220.jpg?w=256)
Calculus
Definite IntegralsIf the function y e x is vertically compressed by a factor of 4 reflected across the x axis and then shifted up 5 units what is the resulting function Write your answer in the form y ceax b Provide your answer below
![Find F x F x F x sin x 6 C 3 t dt](https://media.kunduz.com/media/sug-question-candidate/20230626155459054929-3917327.jpg?w=256)
![Graph the function f x by moving the key points You can only graph integer points for example you cannot plot the point 1 1 Provide your answer below 5 01 0 1 1 2](https://media.kunduz.com/media/sug-question-candidate/20230626154835841897-5066220.jpg?w=256)
Calculus
Definite IntegralsGraph the function f x by moving the key points You can only graph integer points for example you cannot plot the point 1 1 Provide your answer below 5 01 0 1 1 2
![Let f x 3 3 What is f 0 in fraction form Provide your answer below](https://media.kunduz.com/media/sug-question-candidate/20230626154701376025-5066220.jpg?w=256)
![What is the area of the integral of f x 2x 4 within the interval 1 2 10 9 098765 543210 1 2 3 4 5 Created with a trial version of Advan 2x 4 dx 10 o 2x 4 dx 15 O o 2x 4 dx 12 O 2x 4 dx 7 5](https://media.kunduz.com/media/sug-question-candidate/20220622011725853831-3893386.jpg?w=256)
Calculus
Definite IntegralsWhat is the area of the integral of f x 2x 4 within the interval 1 2 10 9 098765 543210 1 2 3 4 5 Created with a trial version of Advan 2x 4 dx 10 o 2x 4 dx 15 O o 2x 4 dx 12 O 2x 4 dx 7 5
![1 Find the area between y x and y x from x 0 to x 3 The aron in](https://media.kunduz.com/media/sug-question-candidate/20230625224608525687-5686493.jpg?w=256)
![rate at which revenue was generated in billions of dollars per year for one large company can be approximated by the equation below where t 5 corresponds to the year 2005 What was the al revenue generated to the nearest billion between the start of 2005 and 2017 f t 1 321 64t 125 5sts 17 www e a definite integral to find the total revenue generated to the nearest billion between the start of 2005 and 2017](https://media.kunduz.com/media/sug-question-candidate/20230625224534545215-5686493.jpg?w=256)
Calculus
Definite Integralsrate at which revenue was generated in billions of dollars per year for one large company can be approximated by the equation below where t 5 corresponds to the year 2005 What was the al revenue generated to the nearest billion between the start of 2005 and 2017 f t 1 321 64t 125 5sts 17 www e a definite integral to find the total revenue generated to the nearest billion between the start of 2005 and 2017
![car leasing firm must decide how much to charge for maintenance on the cars it leases After careful study the firm determines that the rate of maintenance M x on a new car will be approximately M x 45 1 x2 dollars per year where x is the number of years the car has been in use What total maintenance cost can the company expect for a 2 year lease What minimum amount should be added to the monthly lease payments to pay for maintenance on a 2 year lease Write a definite integral to find the total maintenance cost for a 2 year lease dx](https://media.kunduz.com/media/sug-question-candidate/20230625224501605236-5686493.jpg?w=256)
Calculus
Definite Integralscar leasing firm must decide how much to charge for maintenance on the cars it leases After careful study the firm determines that the rate of maintenance M x on a new car will be approximately M x 45 1 x2 dollars per year where x is the number of years the car has been in use What total maintenance cost can the company expect for a 2 year lease What minimum amount should be added to the monthly lease payments to pay for maintenance on a 2 year lease Write a definite integral to find the total maintenance cost for a 2 year lease dx
![Given T 3 a b c f x dx 15 and d nd f 7 g x f x g x dx 7 2g x dx g x g x f x dx 7 3f x dx dx 3 evaluate the followin](https://media.kunduz.com/media/sug-question-candidate/20230625202040607464-3917327.jpg?w=256)
Calculus
Definite IntegralsGiven T 3 a b c f x dx 15 and d nd f 7 g x f x g x dx 7 2g x dx g x g x f x dx 7 3f x dx dx 3 evaluate the followin
![Find F as a function of x and evaluate it at x 0 x 4 and x 2 F x F 0 F 1 F x cos e de RO 11 11](https://media.kunduz.com/media/sug-question-candidate/20230625202336582574-3917327.jpg?w=256)
Calculus
Definite IntegralsFind F as a function of x and evaluate it at x 0 x 4 and x 2 F x F 0 F 1 F x cos e de RO 11 11
![Determine the area of the given region under the curve 1 y](https://media.kunduz.com/media/sug-question-candidate/20230625202236091623-3917327.jpg?w=256)
![Write the limit as a definite integral on the interval a b where c is any point in the ith subinter Limit 4 lim A 0 1 3c 9 Ax dx Interval MY NC 4 5](https://media.kunduz.com/media/sug-question-candidate/20230625201919900760-3917327.jpg?w=256)
Calculus
Definite IntegralsWrite the limit as a definite integral on the interval a b where c is any point in the ith subinter Limit 4 lim A 0 1 3c 9 Ax dx Interval MY NC 4 5
![Write a definite integral that yields the area of the region Do not evaluate the inte f x cos x JO El Ele dx X](https://media.kunduz.com/media/sug-question-candidate/20230625201937063537-3917327.jpg?w=256)
Calculus
Definite IntegralsWrite a definite integral that yields the area of the region Do not evaluate the inte f x cos x JO El Ele dx X
![2 2 3 Points DETAILS a the trapezoidal rule K b Use the trapezoidal rule the midpoint rule and Simpson s rule to approximate the g sin x dx n PREVIOUS ANSWERS the midpoint rule 1 936917 SCALCET9 7 7 008 6](https://media.kunduz.com/media/sug-question-candidate/20230625192141429110-4426144.jpg?w=256)
Calculus
Definite Integrals2 2 3 Points DETAILS a the trapezoidal rule K b Use the trapezoidal rule the midpoint rule and Simpson s rule to approximate the g sin x dx n PREVIOUS ANSWERS the midpoint rule 1 936917 SCALCET9 7 7 008 6
![A chain lying on the ground is 10 m long and its mass is 60 kg How much work in J is required to raise one end of the chain to a height of 4 m Use 9 8 m s for g](https://media.kunduz.com/media/sug-question-candidate/20230625045114632247-4426144.jpg?w=256)
Calculus
Definite IntegralsA chain lying on the ground is 10 m long and its mass is 60 kg How much work in J is required to raise one end of the chain to a height of 4 m Use 9 8 m s for g
![Which of the following formulas would be used to calculate the trapezoid approximation of I x dx using four subintervals 0 T4 1 0 0 2 1 0 5 2 1 1 0 2 1 1 5 1 2 0 O T4 1 0 0 2 1 0 5 2 1 1 0 2 1 1 5 1 2 0 O T4 1 0 0 2 1 0 5 2 1 1 0 2 1 1 5 1 2 0 O T4 1 0 25 2 1 0 75 2 1 1 25 2 1 1 75 23 24 25 26 27 28 29 30 31 32 24 25 26 27 Next](https://media.kunduz.com/media/sug-question-candidate/20220622005722018957-3893386.jpg?w=256)
Calculus
Definite IntegralsWhich of the following formulas would be used to calculate the trapezoid approximation of I x dx using four subintervals 0 T4 1 0 0 2 1 0 5 2 1 1 0 2 1 1 5 1 2 0 O T4 1 0 0 2 1 0 5 2 1 1 0 2 1 1 5 1 2 0 O T4 1 0 0 2 1 0 5 2 1 1 0 2 1 1 5 1 2 0 O T4 1 0 25 2 1 0 75 2 1 1 25 2 1 1 75 23 24 25 26 27 28 29 30 31 32 24 25 26 27 Next
![Find the total area of the shaded regions Ay 5 4 3 2 1 0 14 2 3 3 y e 4 e 5 X Q](https://media.kunduz.com/media/sug-question-candidate/20230624192015377734-5686493.jpg?w=256)
Calculus
Definite IntegralsFind the total area of the shaded regions Ay 5 4 3 2 1 0 14 2 3 3 y e 4 e 5 X Q
![2 2 19 4x 0 9 4x 1 dx Cax 4x 1dx Simplif](https://media.kunduz.com/media/sug-question-candidate/20230624191909057011-5686493.jpg?w=256)
![lim 9c 5 c 4x Write the limit Axl 01 1 int in the th subinterval as a definite integral on the interval 0 5 where c is any](https://media.kunduz.com/media/sug-question-candidate/20230624175744895123-3756363.jpg?w=256)
Calculus
Definite Integralslim 9c 5 c 4x Write the limit Axl 01 1 int in the th subinterval as a definite integral on the interval 0 5 where c is any
![12 Evaluate the following definite integral Note Need the exact answer not a decimal approximation That is answer contains square roots so leave it that way 1 6u 3 du](https://media.kunduz.com/media/sug-question-candidate/20230624170729330018-3756363.jpg?w=256)
Calculus
Definite Integrals12 Evaluate the following definite integral Note Need the exact answer not a decimal approximation That is answer contains square roots so leave it that way 1 6u 3 du
![Find the total area of the shaded regions 13 121 Ay 2 x 1 2 1 X X 13 e](https://media.kunduz.com/media/sug-question-candidate/20230624104600419813-5686493.jpg?w=256)
![d the total area of the shaded regions 16 16 2 Q The total area of the shad Type an integer or a simp](https://media.kunduz.com/media/sug-question-candidate/20230624091639149184-5686493.jpg?w=256)
Calculus
Definite Integralsd the total area of the shaded regions 16 16 2 Q The total area of the shad Type an integer or a simp
![Find the total area of the shaded regions 4 3 2 1 0 14 2 y 2 Q The total area of the shaded regions is Type an integer or decimal rounded to three decimal places as](https://media.kunduz.com/media/sug-question-candidate/20230624091205450701-5686493.jpg?w=256)
Calculus
Definite IntegralsFind the total area of the shaded regions 4 3 2 1 0 14 2 y 2 Q The total area of the shaded regions is Type an integer or decimal rounded to three decimal places as
![Find the total area of the shaded regions 4 3 2 1 0 1 y e Q CO The total area of the shaded regions is Type an integer or decimal rounded to three decimal places a](https://media.kunduz.com/media/sug-question-candidate/20230624081706510621-5686493.jpg?w=256)
Calculus
Definite IntegralsFind the total area of the shaded regions 4 3 2 1 0 1 y e Q CO The total area of the shaded regions is Type an integer or decimal rounded to three decimal places a
![3 15 8x 1dx 15 8x 1 dx Simplify your power](https://media.kunduz.com/media/sug-question-candidate/20230624081615326939-5686493.jpg?w=256)
![Evaluate the definite integral 2 S X 2x 2 O A 0 120 OB 0 001 O C 0 031 C 0 020 dx](https://media.kunduz.com/media/sug-question-candidate/20230624081648367924-5686493.jpg?w=256)
Calculus
Definite IntegralsEvaluate the definite integral 2 S X 2x 2 O A 0 120 OB 0 001 O C 0 031 C 0 020 dx
![Use the Substitution Formula to evaluate the integrals y 4 4 dy and nd 0 5 y 4 dy 38 3 0 Simplify your answer y 4dy S y 4dy](https://media.kunduz.com/media/sug-question-candidate/20230624081134303710-5686493.jpg?w=256)
Calculus
Definite IntegralsUse the Substitution Formula to evaluate the integrals y 4 4 dy and nd 0 5 y 4 dy 38 3 0 Simplify your answer y 4dy S y 4dy
![Find the vertical asimplates of the function Y 2x 1 2 3x 2x x 3 X 2 3 X 0 X 2 3 X 2 3](https://media.kunduz.com/media/sug-question-candidate/20230624063516769625-4832772.jpg?w=256)
Calculus
Definite IntegralsFind the vertical asimplates of the function Y 2x 1 2 3x 2x x 3 X 2 3 X 0 X 2 3 X 2 3
![Find the total area of the shaded regions 4 3 2 1 0 1 2 y e C Q O Points 0 of 1 The total area of the shaded regions is Type an integer or decimal rounded to three decimal places as n](https://media.kunduz.com/media/sug-question-candidate/20230624043428695554-5686493.jpg?w=256)
Calculus
Definite IntegralsFind the total area of the shaded regions 4 3 2 1 0 1 2 y e C Q O Points 0 of 1 The total area of the shaded regions is Type an integer or decimal rounded to three decimal places as n
![Use Simpson s Rule with n 8 to estimate the value of the following integral round your answer to 4 decimal places Le x dx Calculate the upper bound for the error from your calculation How large does n need to be to ensure that the Simpson s Rule estimate is within 0 00005 of the exact value for the above integral](https://media.kunduz.com/media/sug-question-candidate/20230624033220758520-5754634.jpg?w=256)
Calculus
Definite IntegralsUse Simpson s Rule with n 8 to estimate the value of the following integral round your answer to 4 decimal places Le x dx Calculate the upper bound for the error from your calculation How large does n need to be to ensure that the Simpson s Rule estimate is within 0 00005 of the exact value for the above integral
![ind the total area of the shaded regions Q Q The total area of the shaded res Type an integer or a simplified](https://media.kunduz.com/media/sug-question-candidate/20230624043409111109-5686493.jpg?w=256)
Calculus
Definite Integralsind the total area of the shaded regions Q Q The total area of the shaded res Type an integer or a simplified
![Evaluate the definite integral 3 3 1 18 In x 5 X 18 In x 5 X dx dx 1 Round to three decimal placer](https://media.kunduz.com/media/sug-question-candidate/20230624043232933026-5686493.jpg?w=256)
Calculus
Definite IntegralsEvaluate the definite integral 3 3 1 18 In x 5 X 18 In x 5 X dx dx 1 Round to three decimal placer
![Use the Substitution Formula to evaluate the integrals y 4dy and y 4dy 5 y 4dy](https://media.kunduz.com/media/sug-question-candidate/20230624043251693829-5686493.jpg?w=256)
Calculus
Definite IntegralsUse the Substitution Formula to evaluate the integrals y 4dy and y 4dy 5 y 4dy
![Calculate the definite integral 3 3 S 8e5x dx 0 5x](https://media.kunduz.com/media/sug-question-candidate/20230624043159711190-5686493.jpg?w=256)
![Evaluate the definite integral 2 2 3n n 4 dn](https://media.kunduz.com/media/sug-question-candidate/20230624043143450960-5686493.jpg?w=256)
![Use substitution to find the given indefinite integral 5 dx 12 8x](https://media.kunduz.com/media/sug-question-candidate/20230624030735684838-5686493.jpg?w=256)
![Use substitution to find the indefinite integral 8t 3 8 3 3 8 dt 8 dt](https://media.kunduz.com/media/sug-question-candidate/20230624030853752788-5686493.jpg?w=256)
![Find the area between the curve of X O 2 749 O 3 296 O 4 159 O 4 957 w 3 9 3 and the y axis on the interval 2 5](https://media.kunduz.com/media/sug-question-candidate/20220622025910663721-3893386.jpg?w=256)
Calculus
Definite IntegralsFind the area between the curve of X O 2 749 O 3 296 O 4 159 O 4 957 w 3 9 3 and the y axis on the interval 2 5
![Consider the differential equation y xy y 2 Find the general solution and the particular solution passing through the point 0 2 Then plot a slope field and identify the equilibrium solutions Classify each equilibrium as stable or unstable](https://media.kunduz.com/media/sug-question-candidate/20230623022512018748-5750658.jpg?w=256)
Calculus
Definite IntegralsConsider the differential equation y xy y 2 Find the general solution and the particular solution passing through the point 0 2 Then plot a slope field and identify the equilibrium solutions Classify each equilibrium as stable or unstable
![of f 7 For the function f x y sin x 5y find and dx dy of](https://media.kunduz.com/media/sug-question-candidate/20230623035004467874-4709559.jpg?w=256)
![Let 4 x represent the area bounded by the graph the horizontal axis and the vertical lines at and t x for the graph below Evaluate 4 x for x 1 2 3 and 4 4 FFI 2 I 4 3 2 fri 3 4 Q Question Help Video Message instructor Calculator](https://media.kunduz.com/media/sug-question-candidate/20220621111034457099-4642208.jpg?w=256)
Calculus
Definite IntegralsLet 4 x represent the area bounded by the graph the horizontal axis and the vertical lines at and t x for the graph below Evaluate 4 x for x 1 2 3 and 4 4 FFI 2 I 4 3 2 fri 3 4 Q Question Help Video Message instructor Calculator
![Sketch the region enclosed by y e y er and x 1 Find the area of the region](https://media.kunduz.com/media/sug-question-candidate/20220621145452572968-4642208.jpg?w=256)
Calculus
Definite IntegralsSketch the region enclosed by y e y er and x 1 Find the area of the region
![Express 0 0000046 in scientific notation x10 Enter the coefficient that goes in the green box Coefficient](https://media.kunduz.com/media/sug-question-candidate/20220621145254412254-3662668.jpg?w=256)
Calculus
Definite IntegralsExpress 0 0000046 in scientific notation x10 Enter the coefficient that goes in the green box Coefficient