Definite Integrals Questions and Answers

much work is required to move an object from x 2 to x 7 measured in meters with a constant force of 7 N acting along the x axis If a constant force F displaces an object a distance d in the direction of the force what is the correct relationship to use to calculate the work done O A work force distance OB work force distance distance force OD workforce distance OC work

Sketch the following polar rectangle R 1 0 15r54 50s 1 rs4 Choose the correct graph below QA I 3 FO 61 Q B 61 6 X ww O G OC D Q Q OD 6 3 134

Sketch the following region and write an iterated integral of a continuous function f over the region R x y 0 x 2 3x sys 6x 2 x 24 OA 50 OC 3 F Choose the correct iterated integral below OA 2 6x 24 IT 3x P OB f x y dy dx d 50 Q O C OB 50 4 6x 24 OD 6x 24 2 6x 24 Q SO D f x y dx dy f x v dx dv 50 Q

Evaluate the iterated integral J J rcsc0 dr d0 0 2 4

Evaluate the following integral 2 00 0 sin x os y sin z dy dx dz 1 x 2 IT 00 0 Type an exact answer using as needed sin x cos y sin z dy dx dz

Evaluate the following integral as it is written 2x 3 sipx S jx x 4 cos x 2x 3 sip x dy dx dy dx

Let R be the region bounded by the following curves Use the shell method to find the volume of the solid generated when R is revolved about the x axis 2 x 2x 0 and y 10 Set up the integral that gives the volume of the solid Use increasing limits of integration Select the correct choice below and fill in the answer boxes to complete your choice Type an exact answer Sody O A OB BETER dx

Let R be the region bounded by the following curves Use the shell method to find the volume of the solid generated when R is revolved about the y axis y 21 1 21x y 0 x 0 x 1 V Type an exact answer in terms of t GECED O Points 0 of 1 of 26 points 22 yu 21 1 21x

the x axis y x y 0 and x 4 OA dx OB JO by the following curves Use the shell method to find the volume of the solid generated when R is revolved about Set up the integral that gives the volume of the solid Use increasing limits of integration Select the correct choice below and fill in the answer boxes to complete your choice Type an exact answer dy BECCESSO y x R Save

A record club has found that the marginal profit P x in cents is given by P x 0 0005x 0 38x 61 5x for x 700 where x is the number of members currently enrolled in the club Approximate the total profit when 420 members are enrolled by computing the 6 sum P x Ax with Ax 70 i 1 CO The total profit when 420 members are enrolled is approximately Round to the nearest cent as needed Marginal profit in cents AP x 0 0005x 0 38x 61 5x 66 000 52 800 39 600 26 400 13 200 0 0 VV 100 200 300 400 500 600 700 Number of members Q Q

Question 2 The graph of h is given below 6 5 4 3 h x dx h x dx h x dx 6 2 h x dx h x 8 dx 1 6 5 4 3 2 1 Use the graph to evaluate each definite integral dx h ar dz h z dx 2 3 6 1 2 3 6

Find the area of the region bounded by the graphs of y e y e 2 x 1 and x 2 using integration Answers should be exact As a suggestion graph the equations you can see the region for which you are solving for the area Area of region A

h z h x dx such that A expresses the area of the region between the functions y e y e72 1 and z 0 Evaluate the integral to find the area of the region Answers should be exact As a suggestion graph the equations so you can see the region for which you are solving for the area Write an integral in the form A a lower limit of integration 0 b upper limit of integration 0 2 h 2 x 7x 1 Area of region A 0 53279

oint Find the area of the region in the xy plane bounded above by the graph of the function f x 1 6 x 4 3 below by the x axis on the left by the line 8 and on the right by the line 26 The area is

5 Evaluate the definite integral given above Enter your answer as an exact fraction if necessary 51 31 21 8 dt

curves Below we see a region bounded by two curves graph not necessarily to scale The region shaded in light blue is bounded by two curves y 2 x 1 9 in dark red and y 3 x 3 3 in dark blue 150 Part 1 Suppose that we wish to integrate with respect to z to find the value of the shaded area Fill in the blanks so that the resulting integral with respect to will describe the area of the shaded region Part 2 Note Set up the integral so that the lower limit of integration is less than the upper limit of integration Finally after evaluating the integrals above we find that the area of the shaded region equals

Finding the area between curves Below we see a region bounded by two curves graph not necessarily to scale The region shaded in light blue is bounded by two curves y x in dark red and y 2 in dark blue Part 1 Suppose that we wish to integrate with respect to z to find the value of the shaded area Fill in the blanks so that the resulting sum of integrals with respect to 2 will describe the area of the shaded region 68 Note Set up the integral so that the lower limit of integration is less than the upper limit of integration Part 2 Finally after evaluating the integrals above we find that the area of the shaded region equals

3 Let f x x 2x 2x8 x 2 What is the domain of f Is f continuous at 4 Can we define f 4 to make f continuous at 4 If so what is th

Use cylindrical coordinates to find the volume of the region bounded by the plane z 0 and 2 the hyperboloid z 2 x y The volume is Type an exact answer using as needed www ZL

54 4 3 OF 7 wh 5 O 10 3 4 5 6 7 8 O The graph of a function y f x is shown Make a sketch of the antiderivative F given that Fis continuous and F 0 1 3

Let A x 5 4 3 12 2 3 4 5 f t dt with f x shown in the graph below 3 5 At what a value s does A x have a local max x x Use commas to separate multiple values At what a value s does A x have a local min x Use commas to separate multiple values Restrict your responses to values 0 x 6 Ta

Evaluate the following integral in cylindrical coordinates 3 9 x x y 24 S 0 0 0 9 x x y 1 2 x y dz dy dx x y 1 dz dy dx 0 0 0 Type an exact answer using as needed

Find a function F x such that F x 4x 7 and F 2 20 F x

2 While in the barrel of a tennis ball machine the acceleration a in m s of a ball is

Find the sum 4 j 1 2 j

Find the most general antiderivative of the function Check your answer by differentiation Use C for the constant of the antiderivative f x x 3 5x 3 F x 5 3 6x 9x cx

3 6 pts Use the properties of logarithms to fully expand and simplify the expression log 8 x 7 Z 4 6 pts Use the properties of logarithms to write the expression as a single logarithm 3 log x 2 log y 3 log y 5 6 pts Use the properties of logarithms to fully expand and simplify the expression 5 x 1 25 z logs

aluate d 5 5 dx a 5 5 b 2 0 c 2 0

27 Evaluate d 5 5 1 2 x 4 dx x a 5 5 b 2 0 c 2 0

Let A x f t dt where f is the function given by the graph below 2 1 4 3 2 1 1 2 34 4 A 4 1m 2 2 A 7 8 75m 2 Question Help Video 5 X syntax incomplete X syntax incomplete

it is known that f x dx 3 and f x dx 4 find the value of f x dx

The logarithm of a product of two numbers is the same as the sum of the logarithms of these numbers So log4 16 64 log4 16

Suppose f x has the following properties f is continuous and positive for all x 4 14 f is increasing for all x 4 14 11 f a da 112 4 f 4 must be less than

Next question Get a similar question Suppose F 5 9 F 9 4 and F x f x S f x dx 28 X You ca

Suppose we want to evaluate the definite integral Part 1 U Js 5z 9 Re write the definite integral in terms of the variable u and remember to use the limits of integration for the function u f z Then input the antiderivative of the integrand and the limits of integration you found Part 2 de using the substitution u 5x 9

1 Find the most general antiderivative of the function use C as any constant a f x 1 2 2 2 523 b f x 10 29 314 for x 0 c f x 3 for x e f x e for x 0 d f x cos x 5 sin x e f f x 7x 5 8x 4 5 for x 0

b Evaluate the following integrals i ii rin e Sin 1 2 x 2 x 1 d dx x 4 e t e t iii 3x x 2 dx de dt

Solve for 3 14 x You may enter the exact value or round to 4 decimal places

A note card company has found that the marginal cost per card of producing x note cards is given by the function below where C x is the marginal cost in cents per card Find the total cost of producing 980 cards disregarding any fixed costs C x 0 04x 76 for x 1000 The total cost is cents CB

Part 1 Re write the definite integral in terms of the variable u and remember to use the limits of integration for the function u f t Then input the antiderivative of the integrand and the limits of integration you found Part 2 Finally evaluate the original integral by evaluating the antiderivative using limits of integration from Part 1 above

X blem 2 Find an expression for the area under the graph of f x 11x 6 on the interval 5 10 A lim 318 D lim 818 G lim 318 Problem 2 H WI WI WI 557 i 1 2 11 5 si 10 n 10 n 5 5i n 11 5 6 5 10i n 2 11 5 0i 6 5 n 5 10i 5 n 10 6 2 n 10i n 5 5i n 216 11 5 5 6 B lim 8x n H lim n x i 1 n E lim n x n i 1 5 n n 5 5i n 11 i 6 5i n C lim n x 11 5 6 n n 2 10 n F lim n 10i n 2 11 5 10i 6 n n 10 n 5 10i n 2 11 10 6 n

Find exact values or expressions for the six trigonometric functions of angle A Do not use a calculator A 21 29 20 sin A cos A tan A Type integers or simplified fractions csc A sec A cotA

sin 0 2 25 csc Type an integer or a fraction

Use the half angle formulas to determine the exact values of the sine cosine and tangent expressions of the following angle 165 5 sin 165 7 cos 165 5 tan 165 X X Xx

Problem 6 Evaluate the following integrals a 3 2 b 6 S 5 72 dx 0 0 x 1 5x dx

Find the most general antiderivative of the function Check your answer by differentiation Use C for the constant of the antiderivative F x f x 2x4 5x x X 0

Use the properties of logarithms to expand the following logarithm as much as possible Rewrite as a sum difference or product of logs log

Expand In 14xy to rewrite as a sum difference or product of logarithms Choose all correct answers There may be more than one way to rewrite the expression A In 14 In z ln y B 14 ln zy In 14 In ry DC OD 14 ln z 14 ln y E In 2 In z ln y ln 7 F None of the above

Question 10 The graph of g is given below 6 5 4 3 2 1 6 5 4 3 g x dx 1 1 2 3 4 5 6 N 1 3 4 5 6 Use the graph to evaluate each definite integral 2 9 2 9 a dx 9 x dx 9 x dx 9 2 dx 9 x dx a da 19 x 8 dx

Suppose we want to approximate the integral b x dx by using a left hand sum with 4 rectangles of equal widths Write out this sum using function notation for each term Answer Now approximate the integral b a da by using a right hand sum with 3 rectangles of equal widths Write out this sum using function notation for each term Answer