Definite Integrals Questions and Answers

5 The scale on a map is 1 5 cm 1280 m If the distance between the school and the library on the map is 12 centimeters what is the actual distance between the buildings A 15 m B 150 m D C 9600 m D 10 240 m

Find the flux of outward across the curve 9x 4y 36 in the xy plane O 6T O 12T 6T O 12T F 3x y i 2x y i

Suppose F V x y 2 and C is the helical curve parametrised by Evaluate O T O 3T 2 So F F dr I cost y sint 26 05 20 z t t 0

Let S be the portion of the surface z x y with 0 x y and 0 z 9 Which of the following gives the surface area of S O 2 2 TTP 2r 1 r drdo TT 4 r 1 4r drdo T 1 4r drdo 1 27 drde

Let S be the surface parametrised by with 0 u 2 and 0 47 Which of the following gives the surface area of S Sh 2 u dudv O 1 u dudv 4T Luv dude r u v u cos vi usin vj vk u 2 u dudv

Calculate the absolute error in the estimate of of 4 32 da using Simpson s rule with rz 2 Enter an exact value Provide your answer below

If what does the following integral equal 1 y w dw 2 10 3 y w dw 5 and y w dw 6 2 3 Question Help Video Message instructor

The graph below shows a piecewise linear function y f t Use it to answer the questions that follow 5 4 3 2 2 1 0 1 2 3 4 5 6 2 3 4 5 6 7 8 9 Let A z f t f t dt Evaluate A 4 Let B x f t dt Evaluate B 1 Let C z f t dt Evaluate C 9

Find the series representation of 8 x2 2 3 Use your answer to approximate 8 x2 2 3 dx correct to two decimal places

Given that answer 6 f x dx H 29 7 2 f f 6 what is f v du

Evaluate the definite integral and type your complete solution showing each step S 1

70 I In Inx 1 2xcosa x dx Malmsten Integral This might be one of the coolest integrals in calculus it shows the beauty of mathematics I 00 In In x 2rcos a dx where

Evaluate the derivative and type your complete solution showing each step x2 dx sec 1 dr

Plot the point with the given polar coordinates 3 150 O NWA 3 2 1 2 1 0 1 2 3 4 2 1 1 2 2 1 1 2 1 N 2 3 4 X

Find the component form of the indicated vector Let u 5 2 v 2 6 Find 4u 3v O 14 10 O 26 26 26 8 28 32

Integrate the function 2 over the region E that is bounded by the xy plane below and above by the paraboloid 9 82 8y using cylindrical coordinates sqrt 9 8 2pl 2 y adv 0 D 9 81 2 D 5 3 dzdrdo 3 38478pi

47 49 f x x 1 dx J1 2 cos x x3 dx

If a 4 a 3 and an 3an 1 2an 2 then find the value of a6

Which of the following represents the volume of the solid generated by revolving the region bounded by f x x 1 and g x x 1 about the x axis 2x x x 1 x 1 dx 2x x x 1 x 1 dx B 2x x x 1 x 1 dx 2x S x x 1 x 1 dx S x 1 x 1 dx x 1 x 1 dx 1 x 1 x 1 dx 7 x 1 x 1 dx

Here is the graph of 5 cos 3x interval TT A B 3 1 4 C over the TT Give the coordinates of point A the y intercept Give the coordinates of point B the first x intercept along the positive x axis Give the coordinates of point C the first peak of the graph with positive x value

27 f u 2 u 3 du 29 31 4 2 x x w 2 m 6 dx csct cott dr

Part 1 Here is the graph of 3 cos 3x the interval TT A B 7 C 3 1 4 TT Give the coordinates of point A the y intercept 3 0 2 0 2 pi or select in the math editor Give the coordinates of point B the first x intercept along the positive x axis 1 2 0 over To enter type Give the coordinates of point C the first peak of the graph with positive value

2 Use cylinctrical coordinates to evaluate Sff ayzdv E IN here E is the solid in the first octant that lies under the Paraboleid Z 4 2 g y

1 point Solve completely by factoring 8x 5x 4 2x 8x 1 x x x x x 1 11 13 52 50 x x 113 R

Step 3 Therefore the volume of the solid of revolution about the y axis is 243x Exercise b the x axis Step 1 To find the volume of a solid of revolution about the x axis use the horizontal Traver Step 2 Volume V x 50 40 According to the Washer method Volume V V R x rex dx 20 The representative rectangle is shown in the figure The outer and inner radii of this solid are R x 6x and r x 0 respectively V Z 10 So the volume of the solid of revolution about the x axis is 6x 0 dx To find the volume integrate from x dx 243 to x horizontal axis of revolution

9 8 7 6 5 4 3 2 g t 6 8 7 6 5 87 4 3 2 K 2 3935w to 4 5 t 1 2 3 4 5 6 7 8 9 Over which of the given intervals does g have a positive average rate of change

Let g x 76 Use the limit definition of the derivative to differentiate g g x Determine the slope of g at x 6 g 6 Submit All Parts

Professor Ivy wants to buy some honey and has two choices a rectangular bottle that is 5 centimeters by 7 centimeters and 26 centimeters tall or a cylindrical container that has a 1 25 inch radius and is 1 foot tall a Determine the volume of the rectangular container 3 points If needed round the final answer to the nearest tenth do include appropriate units b Determine the volume of the cylindrical container 3 points If needed round the final answer to the nearest tenth do include appropriate units c If the two bottles cost the same amount which is the better buy Explain and provide evidence that justifies your answer if work is on your scratch paper please note that in your answer 3 points

point Evaluate the integral in terms of the constant 6a dr

21 Or s 12 3 Or s 4 9 Or s 4 8 Or s 12 3 FUTA ANC

am stall sells both green tea and mocha ice cream A small portion of either costs 0 75 and large portion costs 1 25 During a short period of time the number of ice creams sold is own in the ta below Green Tea Mocha 0 ii Given that M H 3 4 small 3 Write down a column matrix N representing the cost of each portion of ice cream 3 iii Explain what the numbers given in your answer in ii signify 6 3 6 large 4 evaluate MN 3

Use Stokes Theorem to evaluate F dr where C is oriented counterclockwise as viewed Jc F x y z yzi 3xzj exk C is the circle x y 25 z 4

Question 9 3 points Use integration to find f 4 if d 3x 2 and f 1 5

10 10 Sketch the region whose signed area is represented by the definite integral and evaluate the integral a 2x 4 da

10 9 8 7 6 5 4 3 The focus F is at 1 2 6 S 4 3 2 1 T 2 3 4 S 6 7 8 9 10 O The directrix has equation y 4 2 6 7 8 9 10 0 The vertex V is at 1 1 The line of symmetry has equation 1 symmetry The equation of the parabola is x 1 2 12 y 1 G Enter the equation of the directrix Enter the equation of the line of Enter the equation of the parabola The x intercept s are 2 46 0 4 46 0 there is more than one separate them with commas if none exist enter DNE The y intercept s are 0 0 92 Enter the exact coordinates of the x intercept s if X Enter the exact coordinates of the x intercept s if

Consider the following series n 1 x 1 n n 1 a Use a graphing utility to graph several partial sums of the series A 3 n 3 n 1 K n 2 n 6 n 1 R O 77 n 1 V n 1 n 6 b Find the sum of the series and its radius of convergence 1 x 1 n 2 n 2 B 3 1 d Determine what the approximation represents The sum from part c is an approximation of n 6 n 1 n 6 c Use a graphing utility and 50 terms of the series to approximate the sum when x 0 25 Round your answer to six decimal places 50 1 0 75 n 2 n 2 Determine how good the approximation is Round your answer to six decimal places

Sketch the region whose signed area is represented by the defi nite integral and evaluate the integral a f z 1dx I

6 a Sketch the function f x 2x 2 from x to x 2 b Approximate the signed area of f x on 2 2 by using 4 Is this an overestimate or and right hand sums with n underestimate of the actual signed area c Approximate the signed are of f x on 2 2 by using left hand sums with n 4 Is this an overestimate or and underes timate of the actual signed area d On the graph that you drew in part a sketch the rectan gles that you used in parts b and c 2 to x 2 Find the exact signed area of f x from x

8 If f f x dx 2 and g x dx 3 find a f f x g x dx b f 100 f x dx

Solutions of the differential equation sec y are of the form t tan y True False

2 State the definition of the definite integral in terms of Riemann sums

Which definite integral represents the area of the region that is bounded by 4x 2y2 0 and y x 4 010 4 dy v 4 dy 1 y y 4 dy y 4 dy

If acceleration is a t t 3 and initial velocity is v 0 10 what is an integral function that could be used to find the velocity at any point in time o v t 10 u 3 du f 10 o v t 10 u 3 du o v t 10 u 3 du ov t 10 0 u 3 du 2 3 4 5 6 7 8 9 10 Next

Suppose sin 3x 1 commas 2 2 Find all solutions 0 x 2 Give exact values in radians and type pi for T If there is more than one answer enter the answers separated E

aph y 1 5 cos 3 x 30 0 5 using transformations The f cycles between 0 and 360 U ad 180 y cos x 0 1

Given a b c 0 use Green s theorem find the area of the region D given by ax by c

Consider the given function and the given interval f x 6x 1 x 0 2 a Find the average value fave of f on the given interval fave 65 b Find e such that fave f c Round your answers to three decimal places smaller value larger value A c Sketch the graph of f and a rectangle whose area is the same as the area under the graph of f x f x 6 6

The population of a country from 1950 to 2000 has been modeled with the function P t 60e0 14t where P is measured in millions and t is measured in years with t 0 in the year 1950 What was the average population of the country in the second half of the 20th century Round your answer to the nearest whole number million people

You know that 3 3 fv f t dt 32 3 2 7 7 3 Evaluate 15 Make sure 19 3 3 12 cos x f 6 sin x dx Upload your handwritten work in the box below

3 Given the following matrices if possible determine A If not state Not Possible Answer 11 4 6 9 3 2 6 Selecting a checkbox will replace the entered answer value s with the checkbox value If the checkbox is not selected the entered answer is u