Definite Integrals Questions and Answers

Use the washer method to find the vo Question Find the volume of a solid of revolution formed by rotating the region bounded above by the graph of f x x 4am below by the graph of g over the interval 2 8 about the x axis Enter an exact value in terms of Provide your answer below

Question Consider the region bounded by the graphs of f x 0 7e0 7x g x x 6 and x 10 If this region is rotated about the x axis what is the volume of the resulting solid Select the correct answer below 10 2 7 8 0 760 7x d dx 10 2 O Orf 10 6 0 700 7x 9 81 O 0 49e 4x dx X 8 Ox 0 49e 4x 1 dx dx 81

the washer method to find the volume of a solid of revolution around the y axis Question Let R represent the region bounded on the right by the graph of f y y 2 on the left by the graph of g y and on above and below by y 8 and y 2 What is the volume of the solid of revolution obtained by rotating R about the y axis Enter an exact value in terms of

6 Given the graph of f x find f x dx 7

The region is formed by the graphs of y 2 x and y 8 x Find the volume of the solid obtained by revolving the region about the x axis 512 The volume of the solid obtained by revolving the region about the x axis is EXIB

Let 1 x 15x and let R be the region bounded by the graph off and the x axis on the interval 0 5 a Find the area of the surface generated when the graph off on 0 5 is revolved about the x axis b Find the volume of the solid generated when R is revolved about the x axis a Find the area of the surface 94 2478 square units Type an exact answer using z as needed b Find the volume of the solid 85 87 cubic units Tyne an exact answer using x as needed CIIS

Find the mass of the thin bar with the given density function p x x 10 x for 0 x 3 2 1 The mass of the thin bar is 10 10 5 5 units 3

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 y 338 2x in the first quadrant 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 O A OB S 338 0 dx 2 338 y 2 dy The volume is 54925m Type an exact answer

Starting at the same point on a straight road Anna and Benny begin biking with velocities in mi hr given by vA t 2t 1 and vg t 7 t respectively Complete parts a through c below a Graph the velocity functions for 0 t 7 O A Av 1 16 12 8 44 04 0 1 2 3 4 5 6 7 G OB Av t 16 12 8 mile s and Anna rides mile s and Anna rides mile s 4 0 0 1 2 3 4 5 6 7 O C mile s Anna rides farther mile s Benny rides farther Av t 16 124 8 4 V 0 0 1 2 3 4 5 6 7 OD Av 1 16 12 8 4 0 F 0123456 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 OA Since Benny rides OB Since Benny rides OC Anna and Benny both ride Q

imply the integral that gives the arc length of the following curve on the given interval b If necessary use technology to evaluate or approximate the integral y cos 3x for 0 x OA 1 3 sin 23x dx B T O C 0 T 0 1 9 sin 3x dx R S 0 3 cos 3x dx OD 1 9 cos 3x dx a j 0 b The arc length of the curve is approximately 1 e 3x 9 c

volume of a shifted solid of revolution Question Define R as the region bounded above by the graph of f x x 2 below by the x axis on the left by x 1 and on the right by x 2 What is the volume of the solid obtained by rotating R about the line y 1 Enter an exact value in terms of

1 point Evaluate the integral x x 2 x dx

CURRENT OBJECTIVE Use the washer method to find the volume of a solid of revolution around the y axis Question Find the volume of a solid of revolution formed by rotating the region bounded on the right by the graph of f y y 2 and on the left by the graph of g y over the y axis interval 1 4 about the y axis Enter an exact value in terms of J Provide your answer below

if x 30 y 50 and 1 1 if the slope of the linear regression equation is 0 4 Then the estimated value of y at x 8 is Select one 0 b 4 6 d 6

A solid has a cross section that is a square with side length x where x is the location on the x axis where the cross sectic is located Find the volume of the solid if it extends on the x axis along the interval 0 4 Provide your answer below

Problem 4 Find the volume of the solid of revolution bounded by y 1 x 1 x 0 x 2 and the x axis and rotated around the y axis

OBJECT Use the disk method to find the volume of a solid of revolution around the y axis with a function defined as a function of y Question Define R as the region bounded by the graphs of g y 4y 2y 4 y 1 y 0 and the y axis Using the disk method what is the volume of the solid of revolution generated by rotating R about the y axis Enter your answer in terms of J Provide your answer below

Bonus 1 10th deri f x x 10 10 x

Problem 2 Use the disk method to find the volume of the solid of revolution bounded by the graphs of f x x 2x 3 x 0 x 3 and the x axis and rotated about the x axis

on of an object moving vertically along a line is given by the function s t 4 91 32t 17 F the object over the following intervals b 0 3 c 0 2 d 0 h where h 0 is a real number erage velocity on the interval 0 4 is your answer COLO

Problem 6 Example Determine the area between the curves y x y 2 x and x x20 by integrating a with respect to y b with respect to x

CURRENT OBJECTIVE Determine the volume of a solid by integrating a cross section with a circle or semicircle Question The result of slicing through the center of the solid S is the base described by the ellipse y 1 Cross sections of S perpendicular to the x axis and the elliptical base are circles What is the volume of S Enter your answer in terms of t Provide your answer below

Use the disk method to find the volume of a solid of revolution around the y axis with a function defined as a function of y Question Define R as the region bounded by the graphs of g y 6 y 1 y 2 y 3 and the y axis Using the disk method what is the volume of the solid of revolution generated by rotating R about the y axis Enter your answer in terms of Provide your answer below

3 Explain how to find the value of sin 4200 using the unit circle

The graphs of the quadratic functions f x 6 10x and g x 8 x 2 2 are provided below Observe there are TWO lines simultaneously tangent to both graphs The one easier to imagine is tangent at points near the tops of the parabolas Place a ruler on the screen to see where it is approximately The other one should be on the left of the parabolas It is steeper and has a larger slope In this question you will find their equations You can start this question by setting the points of tangencies as a f a and b g b for example Then try to find equations hint slope involving the unknowns a and b 4 2 10 7 5 F2 5 5 7 5 10 a The line simultaneously tangent to both graphs having the LARGEST slope has equation Two decimal places of accuracy y x b The other line simultaneously tangent to both graphs has equation Two decimal places of accuracy Y x

The graphs of the quadratic functions f x 6 10x and g x 8 x 2 are provided below Observe there are TWO lines simultaneously tangent to both graphs The one easier to imagine is tangent at points near the tops of the parabolas Place a ruler on the screen to see where it is approxima other one should be on the left of the parabolas It is steeper and has a larger slope In this question you will find their equations You can start this question by setting the points of tangencies as a f a and b g b for example Then try to find equations hint slope involving the unknowns a and b 10 7 5 A 2 2 4 2 5 5 7 5 10 a The line simultaneously tangent to both graphs having the LARGEST slope has equation Two decimal places of accuracy y x b The other line simultaneously tangent to both graphs has equation Two decimal places of accuracy Y x

Question The result of slicing through the center of the solid S is the base described by the ellipse 1 Cross sections of S perpendicular to the x axis and the elliptical base are rectangles whose height from the base is one fourth its length What is the volume of S Provide your answer below

a Write the integral that gives the area of the surface generated when the curve is revolved about the given axis b Use a calculator or software to approximate the surface area y 2 cos x for 0 x about the x axis R a Choose the correct answer below x 7 2 2 S 0 x 7 OB 2n S 2 cos x 1 4 sin x dx 0 OA 2n 2 cos x 1 4 sin x dx x 7 OC 2 cos x1 0 2 cos x1 4 sin x dx 7 OD 2 S 2 cos x1 4 cos x dx 0

etermine whether the following statements are true and give an explanation or counter example Complete parts a through d below f b a If the curve y f x on the interval a b is revolved about the y axis the area of the surface generated is f a O A False To obtain the surface area integral of f x when it is rotated about the y axis on a b the function y f x must be solved for x in terms of y This yields the function x g y Then the su f b 2ng y 1 g y dy assuming g y is positive on the interval f a f b f a OB area integral becomes 2n y 1 f y dy b True The surface area integral of f x when it is rotated about the x axis on a b is 2 x 1 f x dy To obtain the surface area of the function when it is rotated about the y axis change a limits of integration to f x evaluated at each endpoint and integrate with respect to y This is assuming f y is positive on the interval f a f b

Find the area of the surface generated when the given curve is revolved about the given axis 1 3 1 y 8x for 0 x about the y axis The surface area is square units Type an exact answer using as needed

nd each of the following unctions f x 3 x g x x 1 a f g State the domain of the function Enter your answer using interval notation b f g State the domain of the function Enter your answer using interval notation c fg State the domain of the function Enter your answer using interval notation d f g State the domain of the function Enter your answer using interval notation

Find if is between 0 and 90 Round your answer to the nearest tenth of a degree 63 cos 80 9770 66 cos 0 5490 69 sin 0 7038 64 sin 0 3971 67 sin 0 9813 70 cos 0 0 9153 65 tan 0 6873 68 tan 8 0 6273 71 cos 0 0 4112

For the following exercises split the region between the two curves into two smaller regions then determine the area by integrating over the x axis Note that you will have two integrals to solve 3 y x and y x x 2 1 YA 5 4 3 2 1 1 g x x x f x x 1 2X

In each triangle below find sin A cos A sin B and cos B 7 8 9 A A 3 1 B C 2 C 1 B 10 A C 4 3 B 2 C B

6 8 pts Answer the following questions about the rational function f x a Write the equations of any vertical asymptotes of the graph of f f x 3 x 5 x 1 x b This function has a horizontal asymptote Write its equation x Y 3 x 5 x 1 x 2x 8 16x 1x 15x c Find the y intercept of the graph of f Write your answer as an ordered pair If this graph doesn t have a y intercept say so 2X 8 x 6x 12x d Find any x intercepts of the graph of f Write your answer s as an ordered pair s If this graph doesn t 15X have any x intercepts say so 12 12 5

Assume f is a nonnegative function with a continuous first derivative on a b The curve y f x on a b is revolved about the x axis Explain how to find the area of the surface that is gen What is the area of the surface generated when the graph of f on the interval a b is revolved about the x axis O A To find the area of the surface that is generated evaluate the integral S 2x x 1 F x dx B To find the area of the surface that is generated evaluate the integral S OC To find the area of the surface that is generated evaluate the integral S OD To find the area of the surface that is generated evaluate the integral S b x 1 x 1 P x dx b 2x 1 f x dx a Sz f x 1 f x dx a

A frustum of a cone is generated by revolving the graph of y 5x on the interval 3 7 about the x axis What is the area of the surface of the frustu The area of the surface is 50x656 square units Type an exact answer using as needed

In calculus it is often desired to replace radial expressions with trigonometric expressions The next group of exercises illustrates this Simplify the given expression as much as possible by making the given substitution Assume 0 0 90 65 Vx 9 x 3 tan 0 66 V 16 x 4 tan 0 67 36 x x 6 sin 0 69 Vx 25 x 5 sec 0 5 71 V4x 25 x tan 0 73 V16 9x x 4 3 sin 8 68 V100x x 10 sin 70 Vx 49 x 7 sec 0 72 9x 1 x tan 8 74 V10 x x 10 sin 8

Add and subtract as indicated Then simplify your answers if possible answers in terms of sin 0 and or cos 0 sin 0 1 13 cose sin 0 15 1 sin 8 17 sin 0 1 cos e 1 cos 8 14 16 cos 8 sin 8 1 cos e 18 cos 0 sin 8 cos e 1 sin 0 1 sin 0

The profile of the cables on a suspension bridge may be modeled by a parabola The central span of the bridge is 1280 m long and 160 m high The parabola y 0 00039x gives a good fit to the shape of the cables where x 640 and x and y are measured in meters Approximate the length of the cables that stretch between the tops of the two towers The length of the cables is approximately Round to the nearest whole number meters 160 m 1280 m

Write each of the following in terms of sin 0 and cos 0 and then simplify if possible Leave all answers in terms of sines and cosines 1 csc 0 cot 8 2 sec 0 cot 0 3 csc tan 0 5 sec 8 csc 8 4 sec tan csc 0 6 csc 8 sec 8

a Write and simplify the integral that gives the arc length of the following curve on the given interval b If necessary use technology to evaluate or approximate the integral 5 y for 5 x 15 a Which of the following is a simplified form of the integral OA OB O C D 15 51 15 615 15 S 51 15 5 5 dx dx 25 25 dx dx b The arc length of the curve is approximately

Find a curve that passes through the point 1 7 and has an arc length on the interval 2 6 given by 2 What is one such curve y 12 2 5 S 6 1 576x dx

a Write and simplify the integral that gives the arc length of the following curve on the given interval b If necessary use technology to evaluate or approximate the integral T 2 y 2 sin x for SXS 3 3 a Set up the integral that gives the arc length of the curve Select the correct choice below and fill in the answer box to complete your choice Type an exact answer OA L OB L 2 SO dy 2 2x JOdx

59 Find the slope of the line y mx It passes through the origin and the point 1 m

6 Find a curve that passes through the point 1 7 and has an arc length on the interval 2 6 given by 1 1 576x d 2 What is one such curve

What differentiable functions have an arc length on the interval a b given by the following integrals Note that the answers are not unique Give all functions that satisfy the conditions a b 1 64x dx a E y 8x C G a Choose the correct answer below Select all that apply A y 8x C C y 32x C y 4x6 15 b b S a C S T 1 64 cos 2x dx B D F OH y 32x C y y 8x5 5 8x5 y C 5 C 4x6 15 C

Suppose the graph of f on the interval a b has length L where f is continuous on a b Evaluate the following integrals in terms of L b 2 1 2x dx a 2 a b b c S f x dx ifc 0 a c

The owner of a photocopy store charges 8 cents per copy for the first 100 copies and 6 cents per copy for each copy exceeding 100 In addition there is a setup fee of 2 30 for each photocopying job a Determine R x the revenue from doing one photocopying job consisting of x copies b If it costs the store owner 3 cents per copy what is the profit from doing one photocopying job consisting of x copies Recall that profit is revenue minus cost a Choose the correct answer below OA O C R x R x 08x 2 30 1 x 100 14x 2 30 100 x 08x 2 30 1 x 100 100 x 06x OB O D R x R x 08x 2 30 1 x 100 06x 4 30 100 x 08x 2 30 1 x 100 06x 2 30 100 x

Find the arc length of the line y 4x 3 on 0 3 using calculus and verify your answer using geometry Which of the following is a simplified form of the integral that gives the arc length of the described line segment 3 OA S 1 4x 0x 4x d dx B S 17 dx c S 3 5 dx OD 1 16x dx 0 The length of the curve is