Definite Integrals Questions and Answers

The following graph of f x consists of line segments and semicircles y f x M N 0 13 Click here for long description Use the graph and the notion of signed area to evaluate the definite integral fi f x dx You get OA 18 4x OB 14 x OC 16 2x OD 14 4x 7 8 9 10 11 12 E 16 2 5x

You re given that 1e dx 51 880 to three decimal places Using properties of definite integrals and exponentials find 2e dx A 224 273 OB 143 931 C Can t tell from the information given OD 2084 075 OE 224 273

Given that f3 dx 63 0 and f3 xdx OB 99 0 OC 324 0 OD 576 A Can t tell from the information given E 117 0 13 5 then 3 2x 3 dx

Find the equation of the sphere of radius 6 centered at 1 5 1 This sphere is given by

Calculate vx w v 1 8 5 w VX W 4 7 1 W

Assume that u v 4 u 5 and v 3 What is the value of 9u u 7v

n limno 2 1 k 1 n 4 O A 2 2x dx O B 2 x dx 4 c 2 x dx 2 O D 2 2x dx O E 2 x dx

Use trapezoids to approximate the area under the curve y f x 5 sin x 2 5 cos 4x on the interval 0 1 using 10 equal subdivisions 04 42 You get Area 0 2 OA 3 157 OB 3 183 C 63 138 OD 3 078 E 6 314 06 08

y f x 5 sin xx 2 5 cos 4xx on the interval 0 1 using 10 equal subdivisions M 04 0 2 You get Area OA 3 196 B 2 780 C 2 078 OD 3 157 C 3 107 0 2 04 06 0 8

Use right hand endpoints and 6 equal subdivisions to approximate the area beneath the curve y f x 1 on the interval 0 6 1 2 0 81 0 61 0 41 0 21 You get Area A 0 9243 B 1 897 C 1 405 D 1 682 1 021

4 A rotating beacon is located 2 miles out in the water Let 4 be the point on the shore that s closest to the beacon As the beacon rotates at 10 revolutions per minute the beam sweeps down the shore once each time it revolves Assume that the shore is straight How fast is the point where the beam hits the shore moving at an instant when the beam is lighting up a point 2 miles downshore from point

The table below gives data points for the continuous function y f x 0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0 f x 30 44 55 68 66 54 39 37 26 25 40 X Approximate the area under the curve y f x on the interval 0 2 using left hand endpoints and 10 equal subdivisions You get Area OA 88 8 OB 454 OC 90 8 OD 96 8 OE 444

2 A rectangular piece of cardboard 8 inches by 14 inches is used to make an open top box by cutting out a small square from each corner and bending up the sides What size square should be cut from each corner for the box to have the maximum volume 8 14

f x 20x3 12x2 6 f 0 3 f 1 0 find f

Find the differential of the function y The differential is dx 12eX X 7

Find the derivative dy y 4 24x 2

You would probably use calculus to determine the area for which of the following shapes OA C OB O C A D E

Evaluate the integral 1 3 1 2 cos 2 cos t i 6 sin t j sec t k dt 1 2 cos t i 6 sin t j sec t k dt i j k Type exact answers

a Evaluate b For what values of a does converge L 1 1 I fe e dr 1 dr

If cos 0 0 and tan 0 0 in which quadrant does 0 lie Select the correct answer below O Quadrant I O Quadrant II O Quadrant III O Quadrant IV

12 Find the particular antiderivative given that f x 15 24 8 and f 1 7 13 The marginal cost of manufacturing x units of a product is given by

Question Provide your answer below 0 in x The figure above shows a right triangle with dimensions in inches If 0 27 what is the length of x to the nearest inch 43 in

1 point Solve the heat problem THINK U Uxx 0 x u 0 1 0 u T 1 0 u x 0 cos x

Find the length of the curve with parametric equations 2t y 3t where 0 t 1 022 1

The function h t 16t 48t 28 models the height in feet of a ball where t is the time in seconds What is the maximum height that the ball reaches Just write the numerical answer do not include h or any units in your answer

Find the y intercept of the following function f x x 13x 42 Write your answer in the form x y and use fractions if necessary

A company is planning to manufacture mountain bikes The fixed monthly cost will be 300 000 and it will cost to produce each bicycle SIXX A Write the cost function C of producing x mountain bikes per month C x B Write the average cost function C of producing x mountain bikes per month C x C Find and interpret C 500 C 1000 C 2000 and C 4000 C 500 Interpret C 500 When bicycles are produced in a month it costs C 1000 Interpret 1000 When bicycles are produced in a month it costs C 2000 Interpret C 2000 When bicycles are produced in a month it costs to produce each bicycle to produce each bicycle to produce each bicycle C 4000 Interpret C 4000 When bicycles are produced in a month it costs to produce each bicycle D What is the horizontal asymptote for the graph of the average cost function C The horizontal asympote is Type an equation Describe what this means in practical terms O A When 300 000 bicycles are produced in a month the cost per bicycle is at a minimum OB When 200 bicycles are produced in a month the cost per bicycle is at a minimum OC The cost per bicycle approaches 200 as more bicycles are produced in a month OD The cost per bicycle approaches 300 000 as more bicycles are produced in a month

A rectangle is constructed with its base on the x axis and its upper two vertices on the parabola y 25 x What are the dimensions of the rectangle with the maximum area What is the area THEONE and the longer The shorter dimension of the rectangle is dimension is Type an exact answer using radicals as needed

If a region is bounded by y lnx x 0 y 2 and y 1 find the definite integral that will compute for the moment with respect to y axis

Consider the parametric equations below x sin t y sin 4t Osts 2 Set up an integral that represents the area of the surface obtained by rotating the given curve about the x a C at dt Use your calculator to find the surface area correct to four decimal places

The table provides various values including select minimums and maximums of a particular cosine function f x Angle T negative 3 times pi over 4 negative pi over 2 negative pi over 4 0 pi over 2 f x 2 1 4 1 2 4 2 What is the period of the function that contains these points

0 24 Shiple points to be left endpoints Round your answer to six decimal places 2006343 What does the Riemann sum represent The Riemann sum represents the net area of the rectangles with respect to the Illustrate with a diagram 5x 00 24 77 8 Select y axis x axis D origin KIT 4 500 3 KIN 00 24 tat t

The graph of the polynomial f x is given below If f x has degree 4 find the factored equation for f x Provide your answer below f x 3 2 1 2 31 2 w 3

a Find the largest possible open interval s on which f is increasing and the largest possible open interval s on which f is decreasing Select the correct choice below and if necessary fill in the answer box es within your choice 13 A The function is increasing on the open interval s 8 12 and decreasing on the open interval s 7 8 Simplify your answer Type your answer in interval notation Use a comma to separate answers as needed B The function is decreasing on the open interval s The function is never increasing Simplify your answer Type your answer in interval OC The function is increasing on the open interval s The function is never decreasing notation Use a comma to separate answers as needed Simplify your answer Type your answer in interval notation Use a comma to separate answers as needed OD The function is never increasing nor decreasing A The critical point s are located at x 8 10 Use a comma to separate answers as needed B There are no critical points b Find the critical points of f Which critical points correspond to local maxima Local minima Neither Find the critical points of f Select the correct choice below and if necessary fill in the answer box within your choice Which critical points correspond to local maxima Select the correct choice below and if necessary fill in the answer box es within your choice A The critical point s x correspond s to local maxima Use a comma to separate answers as needed B There are no critical points that correspond to local maxima

Evaluate the integral using only methods taught in class and simplify your answer Do not skip steps Use either the Finger Method or the Plug and Chug Method to find coefficients when it is possible to do so if you don t no credit will be given x x 1 x 1 x 2 dx

Find fis x dx by using the formula for the area of a triangle 8 8 x dx Simplify your answer

Choose the correct answer below A A level curve is the intersection of the surface with a plane parallel to the xy plane B A level curve is a plot that is made of various traces graphed on one set of axes C A level curve is the intersection of the surface with the yz plane or the xz plane OD A level curve is the trace that results from a surface being cut by a slanted plane

Use the Shell Method to find the volume V of the solid obtained by rotating the region bounded by the following curves about the x axis Draw the 2D region the resulting 3D volume of revolution a typical shell state dV get everything in terms of x or y queen variable and finally integrate to find V Use only methods taught in this class x 3 y 6 x 19

The rate at which revenue was generated in billions of dollars per year for one large company can be approximated by the equation below where t 5 corresponds to the year 2005 What was the total revenue generated to the nearest billion between the start of 2005 and 2017 f t 1 381 2 82t 112 5sts 17 Write a definite integral to find the total revenue generated to the nearest billion between the start of 2005 and 2017 dt

A car leasing firm must decide how much to charge for maintenance on the cars it leases After careful study the firm determines that the rate of maintenance M x on a new car will be approximately M x 53 1 x dollars per year where x is the number of years the car has been in use What total maintenance cost can the company expect for a 2 year lease What minimum amount should be added to the monthly lease payments to pay for maintenance on a 2 year lease Write a definite integral to find the total maintenance cost for a 2 year lease JOdx dx 0

Evaluate the integral using the Trigonometric Substitution method Do this problem in the exact way it was shown in lecture x 81 x 3 dx

Solve the initial value problem explicitly du dx 2 8x 3x 6 and u 3 when x 1

Evaluate the integral using only methods taught in class 2 Tsi sin 0 cos 0d0

Rewrite the integral O O O FIT 0 0 O 2 8 8 p2 0 4 1 8 x 12 x x r dzdrde r dzdrde p sin pdpd qdo 0 0 0 0 p sin und podypde 8 x y dzdxdy in spherical coordinates You do not have to evaluate the integral

Find Kyo for which the dea of the region x 0 and y 0 dave the x axis and below the cure y xx 9 15 9 19 3 2 4

The following graph describes a matrix A as usual that is a colored vector in the domain on left is mapped to same color in the codomain on the right Find two linearly independent eigenvectors and each one s associated eigenvalue 3 y 3 Y 3 2 2 2 3 1 2 X 3 3 2 1 2 1 2 3 1 2 Xx 3

Evaluate the integral 3 b by y dy 0 A 18 b 36 045

Consider the following integral te ts dt t8 Find a substitution to rewrite the integrand as 8 U du dt du Evaluate the given integral Use C for the constant of integra

Consider the following integral x5 1 x6 3 dx Given the substitution u x6 3 find du ox dx du Rewrite the given integral in terms of u JC 1 ou Evaluate the integral by making the given substitution Remember to use absolute values where appropriate Use C for the constant of i

Which of the following is the total length of the curve of intersection of the surfaces x y 2 4 and y 22 4 a 2 3 b 4 3 c 2 d 4 e 8TT