Definite Integrals Questions and Answers

QUESTION 14 3 points The goal in this question is to estimate the value In 1 05 with the help of the Taylor polynomial of order 3 a Find the Taylor polynomial near r 0 for f x ln 1 r So find f 0 f x f 0 f z f 0 f r and f 0 and arrange them in the polynomial b Find the value of r such that f x ln 1 05 desimal places

Show every step of the process for finding the integral below Be sure to but a box around the answer in your hand written work Then indicate the technique s used Select all choices that apply sin x cos xdx O Integration by u substitution O Integration by Parts O Integration by Partial Fraction Decomposition Improper Integral

Given the series below find the following 8 n 2 n 0 a Expand the series showing the first four terms a a a and a in your expansion b Write out the first four partial sums S S S3 and S c Write an expression for S the n th partial sum d Does the series converge or diverge Base your answer on part c not the absolute value of r e If convergent what is S the sum of the entire series Base your answer on the work for part d O Converges

2 Use the Integral Test to determine whether the following series converge after showing that the conditions of the Integral Test are satisfied a b en 1 en 1 k lnk

fr F dr where F x y z yi zjxk and C is a triangle with vertices 0 0 0 Use Stokes theorem to evaluate 9 0 0 and 0 9 9 oriented counterclockwise when viewed from above

Set up and evaluate an integral for the surface area of the surface defined by z x y 1 with 1 z 5

1 Use Simpson s rule with m n 2 to estimate f f x y2 dy dx

Determine the value of the unknown 1 log 3 27 X

A rope passing through a capstan on a dock is attached to a boat offshore The rope is pulled in at a constant rate of 2 ft s and the capstan is 6 ft vertically above the water How fast is the boat traveling when it is 5 ft from the dock Let x be the horizontal distance from the boat to the dock and z be the length of the rope Write an equation relating x and z

At the end of every 3 months Mayan deposits 200 into an account that pays 5 compounded quarterly After 2 years she puts the accumulated amount into a certificate of deposit paying 6 compounded monthly for 1 year When this certificate matures how much will Mayan have accumulated How much interest was earned overall

1 dx Evaluate 1 x by applying Rule a Trapezoidal Simpson s 1 3 Rule b c Simpson s 3 8 Rule

1f ex dx e Use the properties of integrals and 1 zex dxe3e to evaluate 4 ex 5 dx

IF f x dx 35 and g x If g x dx 16 find LAFE 4f x 5g x dx

Write the following expression as a single definite integral of the form m F x f x dx F x dx f x dx f x dx Step 1 f x dx f x dx f x dx Using the first and second integrals since we have 3 1 4 f x dx F x We know that for a b c f x dx f x dx

1 3 Use the Divergence Theorem to evaluate F ds where F x y z z xi First compute integrals over S and S where S is the disk x y2 4 oriented downward and S S U S cos Z sz j x z y k and S is the top half of the sphere x y z 4 Hint Note that S is not a closed

If f is integrable on a b then Use this to evaluate the integral y d Interpret the integral in part c as a difference of areas O The integral represents the area above the x axis minus the area below the x axis O The integral represents the area below the x axis minus the area above the x axis Illustrate with a diagram 10 20 30 n f r x dx 11 dx limf x Ax where Ax n i 1 2 10 20 30 b a 2 and x a iAx 10 20 30 2 4 6 co 8

Draw a dependency diagram and write a chain rule formula for w f x y x m n p y k n p Choose the correct dependency diagram for dw on OA da da du dw n W OA dw Choose the correct chain rule formula for dw dn 316 du dy OB aw dh OC dw dn dy d OD dw an OA ON dw dw ax dy OB dw dy dw dx de dn dy dh dw dx dx an Choose the correct dependency diagram for dwlop OC dw dx Ow dy dx dn dy an da di dx dy an an Choose the correct chain rule formula for dw ap dw dy dy dn dy Op OA ON aw dx do dx dp ow op dw ap ow d de op aw dy dy op dw dw de dy ON dy dx dy ap op Y du ch dy op 316 and 318 given the functions below OB dy ch OB ch 4 CIDE da da Oc du dy OD 316 314 OD dar di di Up W Jav

upper sum lower sum Illustrate 11 FIS 7 3 T DO T 2T FIM KI3 5 3 K 7 O 510 2 T 1 KI3 FI3 5 K IT

With a programmable calculator or a computer it is possible to evaluate the expressions for the sums of areas of approximating rectangles even for large values of n using looping On a TI u Is command or a For EndFor loop on a Casio use Isz on an HP or in BASIC use a FOR NEXT loop Compute the sum of the areas of approximating rectangles using equal subintervals and righ endpoints for n 10 30 50 and 100 n 10 30 50 100 the region under y 4 cos x the x axis and the lines x 0 and x T 2 Sum of Areas Use these estimates to guess the value of the exact area

upper sum lower sum Illustrate TT FIN KIN 2 TT TT KIN 2 FIN 2 TT

Express the integral as a limit of Riemann sums using right endpoints Do not evaluate the limit r6 5 x dx Find the width of each subinterval in terms of n Find the ith endpoint in terms of n x Evaluate f x 5 x at the ith endpoint f x units Express the integral as the limit of Riemann sums using right endpoints lim 318 n i 1

f x 1 cos with n 3 4 and 6 Illustrate each case with a sketch similar to the figure shown below Round your answers to two decimal places n 3 upper sum lower sum Illustrate RS XS b

The graph of the function f is given y 24 18 12 9 0 6 y f x 12 18 24 30 36 a Use six rectangles to find estimates of each type for the area under the given graph of f the x axis and the lines x 0 and 1 L sample points are left endpoints 46 li R sample points are right endpoints R6 X iii M6 sample points are midpoints M

13 u 3i j v 2i 4j Answer a 1 7127 b 98 1

Use the Midpoint Rule with n 4 to approximate the integral 2 7x MA 7x 9 dx

Let f x 9 4x a Estimate the area under the graph of f the x axis and the lines x 1 and x 2 using three rectangles and a right endpoints R Improve your estimate by using six rectangles R6 Sketch the curve and the approximating rectangles for R y 0 1 25 20 15 10 0 1 25 20 15 H 10 5 0 1 25 20 15 10

3g f x 2 cos x 0 sxs 4 evaluate the left Riemann sum with n 6 taking the sample points to be left endpoints Round your answer to six decimal places x What does the Riemann sum represent The Riemann sum represents the net area of the rectangles with respect to the x axis Illustrate with a diagram y 0 24 100 744 3x 8 151 00 24 15100 3 RIN tisto 8 3x 4 08 2 00 4 2 3 15 00 5x 3x

Use Stokes theorem to evaluate F dr F x y z zexi z 6y j x 9z k C is the circle y2 z 9 x 4 oriented clockwise as viewed from

2 Find the average distance from a point P x y in the disk x y 9 to the origin The average distance is 2 Simplify your answer

Find the line integral along the given path C f x y dx where C x t y 9t 2 for 0 st 7 x y dx

The result is calculated as follows 1 x 7 10 dx 107x17 10 1 X 0 F 1 F 0 0

The table below shows the number of accidents fatalities hours flown and miles flown for a country s airlines with scheduled service in selected years Complete parts a and b below Miles flown billions 7 2 7 5 7 7 Year 2005 2008 2012 Accidents Fatalities 53 87 37 22 23 1 Hours Flown millions 17 1 17 5 17 3 SCITE a Compute the accident rate per million flight hours in 2005 and 2012 By this measure has general aviation become safer Select the correct choice below and fill in the answer boxes to complete your choice Round to the nearest tenth as needed OA General aviation has not become safer because the accident rate in 2005 hours OB General aviation has become safer because the accident rate in 2005 accidents per million flight hours is lower than the accident rate in 2012 accidents per million flight hours is higher than the accident rate in 2012 accidents per million flight accidents per million flight hours

1 0 1 Points DETAILS A T Find the area of the region that lies inside the first curve and outside the second curve r 3 cos 8 r 1 cos 8 PREVIOUS ANSWERS x Enhanced Feedback Please try again keeping in mind that you should first find the intersection points Then use the formula for the area between two parametric curves which is A Watch it Ion con dl where 2

DETAILS b sin br 114 0 1 Points Find the solution of the differential equation that satisfies the given initial condition dA Ab cos br A 0 6 dr Need Help x PREVIOUS Read It

Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions r 7 0 0 8 6 4 2 y 8 6 4 2 2 4 6 8F y 8 6 4 2 2 4 6 8 X 8 8 6 4 2 4 6 y 8 6 2 4 2 2 4 6 8F y 8 6 2 2 2 4 4 6 6 8 X 8

DETAILS 11 0 1 Points Find the area of the region that lies inside the first curve and outside the second curve r 3 cos 8 r 1 cos 8 A T x Enhanced Feedback Please try again keeping in mind that you should first find the intersection points Then use the formula for the area between two parametric curves which is A toll

Tutorial Exercise Express the number as a ratio of integers 0 81 0 81818181 Step 1 To write 0 81 as a ratio of integers first write it in expanded form 0 81 as 0 818181 0 81 0 0081 0 000081 Factoring out 0 81 from each of these terms gives the following 0 81 0 81 0 0081 1 0 81 1 0 000081 0 81

Sketch the curve 6 r 5 4 cos 0 4 2 y 2 6 8 2 4 8 6 4 2 y N 2 X

the equation of the sphere in standard form Find the center and radius 43 x y 2 2x 6y 8z 1 0 Answer 44 a y z 9x 2y 10z 19 0 45 9x 9y 92 6x 18y 1 0 Answer x 1 Center 2 y 1 1 1 0 exercises 43 44 45 and 40 complete Radius 1

onsider the following y x y 20 x 10 5 25 20 15 10 10 5 10 X 15 20 a Find the area of the region by integrating with respect to x 243 2 b Find the area of the region by integrating with respect to y

Integrate the given function over the given surface 3 G x y z z over the parabolic cylinder y z 0 x 3 0szs 2 Integrate the function G x y z do S Type an integer or a simplified fraction

100727 MANUE G x y z dg PHILADE PERUMS S 2009 40 Integrate G x y z z over the parabolic cylinder y z 0 x 2 0 zs Write the integral G x y z do as a double integral 777 SPORT dz dx 15M 2

1 2 SSG x y 2 do 2 1 42 S S dzdx S 00 Evaluate the integral found in the previous step SJ G x y z do S

Which of the following functions described does not have an inverse A None of these they all have inverses B Vm the volume of m kilograms of sand OC K t the temperature in Kelvin as a function of the temperature tin Fahrenheit D A d your age as a function of the date d E A t the altitude of an airplane as a function of time t between take off and landing

Use the given transformation to evaluate the integral J R 12x 12y da where R is the parallelogram with vertices 2 4 2 4 4 2 and 0 6 x 1 u v y v2u

Integrate the given function over the given surface G x y z y over the sphere x y 4 Integrate the function SS x y z do S Type an exact answer in terms of T

Find the angle where 0 0 180 between the following vectors Give your answer accurate to at least 2 decimal places u 117 6j 127 137 0

Tolo Co plans the following repurchases 9 6 million in one year nothing in two years and 20 6 million in three years After that it will stop repurchasing and will issue dividends totaling 24 4 million in four years The total paid in dividends is expected to increase by 2 9 per year thereafter If Tolo has 2 3 million shares outstanding and an equity cost of capital of 10 9 what is its price per share today The stock price would be Round to the nearest cent

Daily Enterprises is purchasing a 9 8 million machine It will cost 49 000 to transport and install the machine The machine has a depreciable life of five years and will have no salvage value The machine will generate incremental revenues of 4 2 million per year along with incremental costs of 1 3 million per year If Daily s marginal tax rate is 35 what are the incremental earnings net income associated with the new machine The annual incremental earnings are Round to the nearest dollar

Exercise b If 1 and 2 are the values of r that you found in part a show that every member of the family of functions y ae 1x be2x is also a solution Part 1 of 3 We must decide whether 4F x 2f x 2f x 0 for f x aex 2 be x We have f x and f x