Definite Integrals Questions and Answers

Evaluate the definite integral 2 0 2 cos 0 3 sin 0 de

Use the Evaluation Theorem to decide if the definite integral exists and either evaluate the integral or enter DNE if it does not exist 1 6 6 x x dx

1 1 5 se

Evaluate the definite integral 4 6 2 x 5x dx 0

Evaluate the definite integral 16 1 5 3 2 x dx

The velocity of a fast automobile on a straight highway is given by the function shown where t is measured in seconds and v has units of m s a Graph the velocity function for Osts 100 When is the velocity a maximum When is the velocity zero b What is the distance traveled by the automobile in the first 25 s c What is the distance traveled by the automobile in the first 65 s d What is the position of the automobile when t 80 When is the velocity a maximum Select the correct choice and fill in the answer box to complete your choice A At the specific time s t Use a comma to separate answers as needed B At all values of t in the interval s 20 55 Type your answer in interval notation Use a comma to separate answers as needed When is the velocity zero Select the correct choice and fill in the answer box to complete your choice CELCB A At the specific time s t 0 65 Use a comma to separate answers as needed B At all values of t in the interval s Type your answer in interval notation Use a comma to separate answers as needed b The distance traveled by the automobile in the first 25 s is 600 m Simplify your answer c The distance traveled by the automobile in the first 65 s is 2000 m Simplify your answer 60 m in the positive direction from s 0 2t v t 40 if 0 t 20 if 20 t 55 260 4t 1255

A mass hanging from a spring is set in motion and its ensuing velocity is given by v t 5x cos t for t 0 Assume that the positive direction is upward and s 0 C a Determine the position function for t 0 b Graph the position function on the interval 0 3 c At what times does the mass reach its lowest point the first three times d At what times does the mass reach its highest point the first three times

A simple model for the flow of air in and out of the lungs of a certain mammal is given by the following equation where V t measured in liters is the volume of air in the lungs at time t20 tis measured in seconds and t 0 corresponds to a time at which the lungs are full and exhalation begins Only a fraction of the air in the lungs is exchanged with each breath The amount that is exchanged is called the tidal volume Complete parts a through c below V t H sin 2 a Find the volume function V assuming that V 0 14 L Notice that V changes over time at a known rate V Which equation below correctly gives the volume function t OA V t V 0 V x dx Sv x dx 0 b OC V t V 0 v t dt a OB V 0 V t Sv x dx 0 b OD V 0 V t v t dt

Consider the marginal cost function C x 400 12x 0 03x a Find the additional cost incurred in dollars when production is increased from 150 units to 200 units b Find the additional cost incurred in dollars when production is increased from 400 units to 450 units

If f x 2x 5 and g x 1 a then g f 2 0 3 0 80 01 0 1 0 3 O 80

Consider an object moving along a line with the given velocity v Assume t is time measured in seconds and velocities have units a Determine when the motion is in the positive direction and when it is in the negative direction b Find the displacement over the given interval c Find the distance traveled over the given interval v t 212 16t 30 0 6

Choose the correct graph below The region R is shown in yellow A 18 48 O Points 0 a the region R bounded by the graph of f x 12 x and the x axis on the interval 12 24 Graph and show the region R 48 The area of R is B 48 Ay 48 O C 78 Ar P

the result of integrating a population growth rate between times t a and t b where b a Choose the correct answer OA Integrating a population growth rate results in the net change in the population growth rate from time a to time B Integrating a population growth rate results in the rate of change of the population from time a to time b OC Integrating a population growth rate results in the net change of the population from time a to time b

Let A x be the area of the region bounded by the t axis and the graph of y f t from t 0 to t x Consider the given function and graph f t 8 a Find A 8 b Find A 16 c Find a formula for A x a A 8 Simplify your answer Ay 3 2 1 0 16

0 The function f x logx is transformed into the equation f x log x Identify the parameter and the effect the parameter has on the parent function

Does each equation represent exponential decay or exponential growth Drag and drop the choices into the boxes to correctly complete the table Note If an equation is neither exponential growth nor exponential decay do not drag it to the table Exponential Decay Exponential Growth

roblem 2 Evaluate the following integral x 5 x 6x dx

Problem 6 Use the Comparison Theorem to determine which of the following integrals converge S Problem 6 arctanx Spo 2 201 dx Select dx 0 CT 2 2 sinx x512 dx A ii and iii only B i and iii only C i and ii only D i only E none of them F 1 only G iii only H all of them

Problem 5 Evaluate the following improper integral el x x2 Problem 5 So dx If the integral does not converge then write divergent Enter your answer symbolically as in these examples If the integral does not then write divergent

Problem 5 Evaluate the following sum 43 1 2 i 1 i 1 i 2 1

Problem 2 Evaluate the following integral 4 1 x 5 x 6x dx

em 1 Suppose that a sequence is defined by a 1 an 1 1 a 8 To show that a is monotonic using mathematical induction which of the following would be the second step A Assume ax 1 8 x 2 8 and show that a ax 1 B Assume a 2 8 ax 1 8 and show that az 1 ak C Assume ax 1 a and show that a 2 8 x 1 8 D Assume a ax 1 and show that a 8 ax 1 8 E Assume ax 1 8 a 8 and show that ax 1 ax F Assume a 8 ax 18 and show that a ax 1 G Assume a ak 1 and show that a 1 8 x 2 8 H Assume a 1 a and show that a 1 8 x 8

Find the extremals for the following functional f y 2y dx

Match the following expressions that have the same trig values cot 180 cos 3 sin 210 tan 31 4 sin 150 cos 211 3 sin 270 tan 31 2

x uI X

sin In x dx

Which equation describes the circle with center 5 1 and radius 7 a x 5 2 y 1 7 b x 5 2 y c x 5 2 y 1 7 d x 5 2 y 1 49 1 49

sin x 3 cos 2 S dx e 4 f In x x dx

1 fxe dx

TT 2 2x cos x dx 7

2 So 2x 1 x dx

6 sin x x dx

P I zx S G

7 fi 2x e da

3 e ln x da

c 92 S S s sino sin 0 5 cos do

u b S sto 5 x 2X fxsts T c S 5 sine XP XP 5 cus de

a trung integral berikut 4 b 2 sx s x 2 te 3 dx dx 5 sme scuso do

Problem 31 1 point Compute the integral f 2 sin x cos x dx A 3 5 B C D 4E 2 24 7 15 Problem 32 1 point Compute the integral f 3 cos x cos 5x dx 3 A 2 B 2 C 3 D 2 2 E V3 16 20 12

f t dt 8 3 f t dt sf t 3 dt f f t 33d 1381 3 3 4 2 26 f t dt dt 4

3 12 points In the following problem the function f x defined for x 1 is given by the graph below 1 1 2 1 b 3 points Evaluate 2 a 3 points Evaluate T 6f x dx 3 f x 2 dr 4 5 6 7 C 3 points Choose a value of b that would make the definite integral I d 3 points Compute the average value of f x from x 2 to x 6 8 f x dx negative

the following definite and indefinite integrals 14 pts a f x 1 2x 3 dx b 2e 1 dx Tr

use geometry Sax d dx a 10 b S 4x 1 dx to evaluate the definite integral

In a psychology experiment rats were placed in a T maze and the proportion of rats who required more than t seconds to reach the end was as shown below S 0 09e 0 09s ds Use this formula to find the proportion of rats who required more than 9 seconds Round your answer to one decimal place

Solve the differential equation and initial condition and verify that your answer satisfies both the differential equation and the initial condition y v y 6x5 Ly 0 1

Use integration by parts to find the integral Hint Take u x dv xex and use a substitution to find v from dv Use C for th constant of integration x ex dx

1 ola 19 If f a Jo VB 2dt dt which of the following is FALSE HINT The integrand is positive when defined OFONTE DEL SI elsy Dini odt buidat B f is continuous at x for all x 0 E f 1 0 A f 0 0 1 C f 1 0 D f 1 3

In ln x lima In 2

4 where Assume that a particle s position on the x axis is given by x 3 cost 4 sint x is measured in feet and t is measured in seconds a Find the particle s position when t 0 t and t 2

The total accumulated costs C t and revenues R t in thousands of dollars respectively for a photocopying machine satisfy 1 C t 58 and R t 7t8 e t 56 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 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