# Definite Integrals Questions and Answers

Calculus

Definite IntegralsCompute the following derivative using the method of your choi cos x5 sin 5x dx cos x5 sin 5x

Calculus

Definite IntegralsFind the arc length of the curve below on the given interval y In x x 1 for 1 x 65 Hint Integrate with respect to y The length of the curve is Simplify your answer

Calculus

Definite IntegralsUse the inverse relations between In x and e exp x and the properties of In x to prove the following properties a exp 0 1 b exp x y exp x exp y c exp x exp xy y rational a How can exp 0 be calculated Choose the correct answer below OA Find In exp 0 B Calculate exp 0 as the product of a constant and 0 c Find a value a for which In a 0 and substitute In a for 0 D Calculate the limit of exp x as x approaches 0 X S dt Recall that Inx X dt For what value is Type an exac answer X S dt 0

Calculus

Definite IntegralsFind the volume of the solid generated when R shaded region is revolved about the given line 1 x y 1 x 1 about y 1 The volume of the solid obtained by revolving the region about 1 i 5x

Calculus

Definite IntegralsDetermine the following indefinite integral Check your work by differentiation S5 dr dr

Calculus

Definite IntegralsFind the following integral Note that you can check your answer by differentiation S 7 e2z dx

Calculus

Definite IntegralsFactor completely using synthetic division 3x 5x 4x 4 given that 2 is a root x intercept value Write the answer in linear factor form You must show work to receive credit

Calculus

Definite IntegralsProblem 5 Use the method of cylindrical shells to find the volume when the region bounded above by x y and x 2 y and below by the x axis and rotated around the x axis

Calculus

Definite IntegralsFind the volume of the solid generated when R shaded region is revolved about the given line y 1 x y 1 x 1 about y 1

Calculus

Definite IntegralsFor the region bounded by y 2x the x axis and x 2 determine which of the following is greater the volume of the solid generated when the region is revolved about the x axis or about the y axis 32T 3 When the region is revolved about the x axis the volume is Type an exact answer using as needed 16x When the region is revolved about the y axis the volume is 3 Type an exact answer using as needed

Calculus

Definite IntegralsConstruct a circle with radius r Then construct two hexagons H1 and H2 so that the circle is inscribed in H2 and encapsulating H1 H2 has apothem r and H1 has side r Set r 10 and calculate an upper and lower bound for the area of the circle by calculating the difference in the areas of H1 and H2

Calculus

Definite IntegralsAccording to the integral estimate what is the smallest number of terms of the series n 1 be added in order to estimate its sum with an error that is less than 0001 that would have to

Calculus

Definite IntegralsSolve each of the following problems In each case be sure to make a diagram of the situation with all the given information labeled 1 Isosceles Triangle The two equal sides of an isosceles triangle are each 42 centimeters If the base measures 30 centimeters find the height and the measure of the two equal angles Round your answer to the nearest tenth 2 Equilateral Triangle An equilateral triangle one with all sides the same length has an altitude of 4 3 inches Find the length of the sides 3 Length of an Escalator How long should an escalator be if it is to make an angle of 33 with the floor and carry people a vertical distance of 21 feet between floors Round to the nearest tenth 4 Height of a Hill A road up a hill makes an angle of 5 with the horizontal If the road from the bottom of the hill to the top of the hill is 2 5 miles long how high is the hill 5 Circus Tent A rope from the top of a circus tent pole is 72 5 feet long and is anchored to the ground 43 2 feet from the bottom of the pole What angle does the rope make with the pole Give your answer to the nearest tenth of a degree

Calculus

Definite IntegralsQuestion Use the washer method to calculate the volume of the solid formed when the region bounded by y 2x and y 2 5 revolved around the the line y 3 Provide your answer below

Calculus

Definite IntegralsUse the washer method to find the volume of a shifted solid of revolution Question Find the volume of a solid of revolution formed by revolving the region bounded above by the graph of f x 3x 1 an below by the x axis over the interval 5 6 about the line y 4 Enter an exact value in terms of Provide your answer below

Calculus

Definite IntegralsY TO 8 10 12 14 16 Years after 1980 G ue curve shows national health expenditures in billions of dollars The red curve shows its derivative Next question National health expenditures in billions of dollars from 1980 to 1994 are given by the function f t in the graph to the left a How much money was being spent at the beginning of 1983 700 billion b Approximately how fast was the rate of spending increasing in 1990 billion per year

Calculus

Definite IntegralsUse a graphing utility to graph the polar equation inner loop of r 8 12 sin 0 Find the area of the given region Round your answer to four decimal places al Enter a number

Calculus

Definite Integralsa Use the definition of derivative limit process to find f x of the function f x x 2x b Use the result of a to find the equation of the normal line to f at x 1

Calculus

Definite IntegralsThe region bounded by the graphs of x 0 x Iny and x 7 6 Iny in the first quadrant is revolved about the y axis What is the volume of the resulting soll The volume is Type an exact answer

Calculus

Definite IntegralsLet R be the region bounded by the following curves Let S be the solid generated when R is revolved about the given axis If possible find the volume of S by both the disk washer and shell methods Check that your results agree y O A 1 x 1 B X y 1 revolved about the x axis Set up the integral s that gives the volume of the solid as a single integral if possible using the disk washer method Use increasing limits of integration Select the correct choice below and fill in any answer boxes within your choice dx Type exact answers JOdx dx Type exact answers

Calculus

Definite IntegralsThe region is formed by the graphs of y 3 2x and y 8 x Find the volume of the solid obtained by revolving the region about the x axis ronion about the x axis is CLE

Calculus

Definite Integralshe region is formed by the graphs of y 3 x and y 4 x Find the volume of the solid obtained by revolving the region about the x axis The volume of the solid obtained by revolving the region about the x axis is PREURE

Calculus

Definite IntegralsThe region is formed by the graphs of y 3 2x and y 8 x Find the volume of the solid obtained by revolving the region about the x axis The volume of the solid obtained by revolving the region about the x axis is E

Calculus

Definite IntegralsFind the length of the following curve 2 y In x x 1 for 10 145 The length of the curve is Type an integer or decimal rounded to three decimal places as needed

Calculus

Definite IntegralsFind the mass of the thin bar with the given density function p x x 8 x for 0 x 2 The mass of the thin bar is Gyne an exant porwer units

Calculus

Definite IntegralsWrite and simplify the integral that gives the arc length of the following curve on the given interval Then use technology to approximate the integra x 2 4 on 3 9 The integral that gives the arc length of the curve is L JOdx

Calculus

Definite IntegralsFind the area of the shaded region shown in the figure which is not shown to scale

Calculus

Definite IntegralsStarting at the same point on a straight road Anna and Benny begin biking with velocities in mi hr given by VA t 2t 2 and Vg t 5 t respectively Complete parts a through c below a Graph the velocity functions for Osts5 OA Av 0 16 12 8 4 04 0 U 2 3 5 B Av t 16 12 8 4 0 mile s and Anna rides VU mile s and Anna rides VA 2 3 4 5 mile s Benny rides farther mile s Anna rides farther C Av t 16 12 8 4 0 0 1 2 3 4 Q OD b If the bikers ride for 1 hour who rides farther Interpret your conclusion geometrically using the graph in part a Select the correct choice and fill in the answer box es to complete your choice Type integers or decimals A Since Benny rides B Anna and Benny both ride 4 mile s OC Since Benny rides Av 1 16 12 8 4 0 3 4 Q Q

Calculus

Definite IntegralsAnswer the questions about the work of a spring a It takes 70 J of work to stretch a spring 0 2 m from its equilibrium position How much work is needed to stretch it an additional 0 4 m b It takes 70 N of force to stretch a spring 0 2 m from its equilibrium position How much work is needed to stretch it an additional 0 4 What is the work done by a variable force F moving an object along a line from x a to x b in the direction of the force b OA W xF x dx SxF x dx a OB W xF x O c W F x D W F x dx F x dx a The work required to stretch the spring an additional 0 4 meter is 560 joules Type an integer or a decimal b The work required to stretch the spring an additional 0 4 meter is 210 joules Tung an integer or a decimal

Calculus

Definite IntegralsQuestion Consider the region bounded by the graphs of y 7x y ln x y 6 and the x axis If this region is rotated about the y axis what is the volume of the resulting solid Select the correct answer below 6 Of y e dy Oxf 4 46 dy Oxf et y dy J

Calculus

Definite IntegralsQuestion Consider the curve f y y Jour for 1 y 2 Find the arc length of the curve 4y

Calculus

Definite IntegralsQuestion Consider the curve f y 6y 1 over the interval 2 y 6 Which of the following integrals represents the arc length of the graph of f y Select the correct answer below O 6 24y 1 dy 6 6y 1 dy 576y 1 dy 6y 2 dy 6y 1 1 dy

Calculus

Definite IntegralsConvert the following degree measure to radians 90 90 radians Simplify your answer Type an exact answer u

Calculus

Definite IntegralsQuestion Consider the curve f x x 1 for 1 x 4 Find the arc length of the curve 2x Enter an exact answer Provide your answer below

Calculus

Definite IntegralsFind the mass of the following thin bar A 5 m bar with a density in g m of p x 250c For a thin bar or wire represented by the interval a sxsb with a density function p with units of mass per length what is the mass of the object d OA m P x dx OB m m b Sp x dx 8 d dx P x OC m for 0 x 5 b OD m xp x d Sxp x dx

Calculus

Definite IntegralsFind the following lim cos 7a sin 0 0 cos 72 cos sin 72 sin cos 72 sin 7x sin cos 7x cos cos 72 sin sin 72 cos cos 72 sin sin 72 COS Consequently cos 72 sin 72 cos 7x cos 71 b sin 7x 8 Part 2 Part 3 o Part 4 of

Calculus

Definite IntegralsCURRENT OBJECTIVE Use the washer method to find the volume of a solid of revolution around the x axis Question Let R represent the region bounded above by the graph of f x 4x 1 below by the graph of g x 2 and on each side by x 1 and x 4 What is the volume of the solid of revolution obtained by rotating R about the x axis Enter an exact value in terms of dow

Calculus

Definite IntegralsQuestion Consider the curve f y y for 1 y 4 Find the arc length of the curv

Calculus

Definite IntegralsQuestion Find the volume of a solid of revolution formed by revolving the region bounded by the graphs of f x xand g x 1 over the interval 1 3 around the x axis Hint graph the functions to determine which graph forms the upper bound and which graph forms the lower bound then use the washer method Provide your answer below

Calculus

Definite IntegralsQuestion 1 Consider the curve f x x for 1 x 2 Find the arc length of the curve 2x Provide your answer below

Calculus

Definite IntegralsProceeds from Notes Payable On January 26 Bella Co borrowed cash from Conrad Bank by issuing a 120 day note with a face amount of 72 000 Assume a 360 day year a Determine the proceeds of the note assuming the note carries an interest rate of 8 b Determine the proceeds of the note assuming the note is discounted at 8 A