Definite Integrals Questions and Answers

1 Find the area between y 3x and y x from x 0 to x 2 The area is

Find 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

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

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

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

3 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

Find F x F x F x sin x 6 C 3 t dt

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

Let f x 3 3 What is f 0 in fraction form Provide your answer below

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

1 Find the area between y x and y x from x 0 to x 3 The aron in

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

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

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

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

Determine the area of the given region under the curve 1 y

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

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

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

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

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

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

2 2 19 4x 0 9 4x 1 dx Cax 4x 1dx Simplif

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

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

Find the total area of the shaded regions 13 121 Ay 2 x 1 2 1 X X 13 e

d 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

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

3 15 8x 1dx 15 8x 1 dx Simplify your power

Evaluate 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

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

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

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

ind 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

Use 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

Evaluate the definite integral 2 2 3n n 4 dn

Use substitution to find the given indefinite integral 5 dx 12 8x

Use substitution to find the indefinite integral 8t 3 8 3 3 8 dt 8 dt

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

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

of f 7 For the function f x y sin x 5y find and dx dy of

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

Sketch 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