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# CALC ANALYT GEOM II MATH 125

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This 62 page Class Notes was uploaded by Addison Beer on Wednesday September 9, 2015. The Class Notes belongs to MATH 125 at University of Washington taught by Daniel Pollack in Fall. Since its upload, it has received 13 views. For similar materials see /class/192073/math-125-university-of-washington in Mathematics (M) at University of Washington.

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Date Created: 09/09/15

January 30 2008 Announcements o This Week Finish 62 63 Midterm 1 review and 64 o Homework 3 Week 3 Problems Due Tomorrow Thursday January 31 Covers 61 62 and 63 see web for assignment 0 Midterm 1 Tomorrow Thursday January 31 Covers all material in sections 410 51 55 and 61 63 No graphing calculators yes scientific One 85 x 11 handwritted sheet of notes Do sample midterms from Math 125 Materials webbage click on Week 4 MIDTERM 1 Give Exact Answers 0 Sasha Aravkin review session TODAY 500 700 PM in DEN 311 Today 0 62 amp 63 Volumes of Solids of Revolution Summary and Review Problems 0 Additional Review Problems To compute the volume of a solid 8 of revolution 1 Sketch the region to be rotated and indi cate the axis of revolution 2 Decide which method to use slices or shells 3 Determine whether you have an integral with respect to 1 or with respect to y 4 Express either the area of the cross sec tion disk or washer or the area of the cylindrical shell in terms of the variable of integration 5 Determine the bounds of integration 6 Write down the correct definite integral 7 Evaluate the integral to find the volume 2 To keep in mind when computing the volume of a solid of revolution 0 The general principle is that Volume is the Integral of Area 0 In general cross sections yielding disks or washers should be perpendicular to the axis of rotation If you rotate with respect to a vertical axis the cross sections are horizontal and the variable of integration is y If you rotate with respect to a horizontal axis the cross sections are vertical and the variable of integration IS 3 o In general the axis of the cylindrical shells coincides with the axis of rotation The shells have radius equal to the distance to the axis The typical cylindrical shell is generated by slic ing the region parallel to the axis of rotation and rotating the resulting line segment If you rotate with respect to a vertical axis the radius is expressed in terms of x and the variable of integration is x If you rotate with respect to a horizontal axis the radius is expressed in terms of y and the variable of integration is y Example Problem 20 63 Find the volume of the solid obtained by rotating the region bounded by y x2 and 1 y2 about y 1 y y Chapter 6 Review problem 16 Let R be the region in the first quadrant bounded by the curves y x3 and y 21 2 a Find the volume of the solid obtained by rotating R about the x axis b Find the volume of the solid obtained by rotating R about the y axis April 16 2007 Announcements o This Week 64 Midterm Review 65 o Homework 3 Week 3 Problems Covers 61 62 and 63 see web for assignment Due Tomorrow Tuesday April 17 Hand in Thursday April 19 in Section 0 Midterm 1 This Thursday April 19 Covers all material in sections 410 51 55 and 61 63 No graphing calculators yes scientific One 85 x 11 handwritted sheet of notes Do sample midterms from Math 125 Materials webpage click on Week 4 MIDTERM 1 Today 0 62 amp 63 Volumes of Solids of Revolution Summary and Review Problem 0 364 Work To compute the volume of a solid 8 of revolution 1 Sketch the region to be rotated and indi cate the axis of revolution 2 Decide which method to use slices or shells 3 Determine whether you have an integral with respect to 1 or with respect to y 4 Express either the area of the cross sec tion disk or washer or the area of the cylindrical shell in terms of the variable of integration 5 Determine the bounds of integration 6 Write down the correct definite integral 7 Evaluate the integral to find the volume 2 To keep in mind when computing the volume of a solid of revolution 0 The general principle is that Volume is the Integral of Area 0 In general cross sections yielding disks or washers should be perpendicular to the axis of rotation If you rotate with respect to a vertical axis the cross sections are horizontal and the variable of integration is y If you rotate with respect to a horizontal axis the cross sections are vertical and the variable of integration IS 3 o In general the axis of the cylindrical shells coincides with the axis of rotation The shells have radius equal to the distance to the axis The typical cylindrical shell is generated by slic ing the region parallel to the axis of rotation and rotating the resulting line segment If you rotate with respect to a vertical axis the radius is expressed in terms of x and the variable of integration is x If you rotate with respect to a horizontal axis the radius is expressed in terms of y and the variable of integration is y Chapter 6 Review problem 16 Let R be the region in the first quadrant bounded by the curves y x3 and y 21 2 a Find the volume of the solid obtained by rotating R about the x axis b Find the volume of the solid obtained by rotating R about the y axis If an object moves along a straight line with position 50 then the force F on the object in the same direction is defined by Newton s Second Law of Motion d2s F m Force 2 mass gtlt acceleration Newton 2 kg gtlt in2 When the acceleration is constant so is the force F We then define the work to be the product of the force F and the distance d that the object moves W Fd work force gtlt distance Joule 2 Newton gtlt meter 0 Careful Weight is a measurement of force so there is no need to multiply it by g the acceleration due to gravity Suppose that an object moves along the 1 axis in the positive direction from 1 a to 1 b and that at each point 1 between a and b a force f13 acts on the object where f13 is a continuous function To estimate the work done we divide a b into n subintervals with end points x0 and equal width Ax Choose sample points xi 6 x 113 For large n Ax is small and the values of f in x 113 are very close to fxff The work Wi that is done in moving the particle from 141 to xi is approximately Thus the total work done in moving the parti cle from a to b is approximately W N in fa Aa i1 Letting n gt 00 we have Wzabfadx Hooke s law The force required to maintain a spring stretched 1 units beyond its natural length is proportional to 13 fx kw k gt O is the spring constant Example 1 64 8 A spring has natural length 20cm If a 25N force is required to keep it stretched to a length of 30cm how much work is required to stretch it from 20cm to 25cm March 12 2008 Announcements o This Week Finish 93 94 and Final Exam Review 0 Homework 9 Week 9 Problems Due this Week will not be collected Problems 1 2 and 4 from 91 amp 93 should be done by Tuesday 0 Final Exam This Saturday from 130 420 pm in AND 223 Instruction Page available on the web page Cumulative Exam Covers all material No graphing calculators yes scientific One 85 x 11 sheet of notes both sides May use any of the 20 integrals from the table on p 506 in 75 of the text without deriving them Show your work in evaluating any other integrals even if they are on your note sheet Do sample Finals Math 125 Materials webpage Review Session Thursday 530 730 pm in THO 135 Instruction Page available on the web page Today 0 94 Exponential Growth and Decay Coffee Consumption and Newton s Law of Cooling 0 Begin Final Exam Review Coffee consumption Evidence shows that caffeine in the blood is metabolized at a rate proportional to the amount present UW researchers find that typically the amount of caffeine in the blood is reduced by 12 in 1 hour A typical espresso from the HUB has approx imately 50 mg of caffeine Problem Suppose you have an espresso at time t 0 just before your calculus lecture a At time 75 how much caffeine is in your blood b How much caffeine is in your blood 5 hours later when you go to the Math Study Center Newton s Law of Cooling Newton s Law of Cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings provided that this difference is not too large Let Tt be the temperature of the object at time 25 let T5 be the temperature of the sur roundings Newton s Law of Cooling formu lated as a differential equation states dT k T T dt s where k is a negative constant If TO To then Tot 2 T8 To me 94 Problem 14 A thermometer is taken from a room where the temperature is 200C to the outdoors where the temperature is 50C After one minute the thermometer reads 120C 0 What will the reading on the thermometer be after one more minute c When will the thermometer read 60C TODiCS 120 review Antiderivatives Areas and distances Riemann sums definite inte grals fundamental theorem of calculus total change Indefinite integrals net change theorem Techniques of integration 0 Substitution rule o Integration by parts o Trigonometric substitution trigonometric inte grals Partial fractions includes long division of poly nomiIas Rationalizing substitutions Approximate integration Midpoint Trapezoid and Simpson s rule Improper integrals includes I Hospital s rule Infi nite intervals discontinuous functions 7 Applications of integration Area between curves includes graphing func tions Volumes cross section and shell methods Work Average value of a function Arc length Moments and centers of mass 8 Differential equations modeling and solving 125 Final Exam Winter 2005 Problem 1 Evaluate the following integrals 2 2 0 51336 da 0 COSx o nsmxdx snx o 12005da da V5132 213 3 x44 d3 513 2132 Problem 2 o For b positive evaluate the integral b da 0 132 81 12 Simplify your answer o Evaluate the improper integral 00 da 0 132 81 12 7 or if it doesn t converge explain why Problem 3 Find the area of the region bounded by the curves y cos x y x2 1 1 O and 13 2 2 Problem 4 Let ft1t Vl I s4ds and let 933 21xftdt Find gquot2 Problem 5 Let b be a positive number and consider the region bounded by the curves y 5132 yx21 332 19 and 5131 1 Find the y coordinate of the center of mass of this region in terms of b 2 Because of the symmetry of this region the x coordinate of the center of mass is O For small values of b say b lt M the center of mass is in the region while for b gt M the center of mass is outside of the region Find M Problem 6 Consider the region bounded by yxSin7rx y 0 and 13 1 1 Set up an integral for the volume of the solid obtained by rotating this region about the y axis DO NOT EVALUATE 2 Set up an integral for the volume of the solid obtained by rotating this region about the x axis DO NOT EVALUATE 10 Problem 7 A turkey has an internal tem perature of 60 degrees F At 1200 pm it is placed in an oven whose temperature is at 380 degrees 1 Newton s Law of Cooling states the rate of cooling or heating of an object is pro portional to the temperature difference be tween the object and its surroundings Use Newton s Law of Cooling to write down a differential equation and initial condition for the internal temperature y of the turkey at time 25 Take 75 O to be the time it is placed in the oven Your equation should involve an unknown constant k 2 After 2 hours it is observed that the tem perature of the turkey is 140 degrees Use this to find the constant k and the temper ature y as a function of time t 3 At what time will the temperature equal 170 degrees For this problem give a dec imal answer 11 Problem 8 The region between the curve y f13 the line 1 1 and the line y 4 is rotated around the line 13 1 A formula for f13 is not known however we do have the following a table of values 1 115225 3 354 f13 o 3 7 12 19 28 4 Use Simpson s Rule to approximate the volume of this solid of revolution For this problem give a decimal answer 12 March 14 2008 Announcements o This Week Finish 93 94 and Final Exam Review 0 Homework 9 Week 9 Problems Due this Week will not be collected 0 Final Exam This Saturday from 130 420 pm in AND 223 Instruction Page available on the web page Cumulative Exam Covers all material No graphing calculators yes scientific One 85 x 11 sheet of notes both sides May use any of the 20 integrals from the table on p 506 in 75 of the text without deriving them Show your work in evaluating any other integrals even if they are on your note sheet Do sample Finals Math 125 Materials webpage Review Session Thursday 530 730 pm in THO 135 Today 0 Final Exam Review Winter 2005 TODiCS 120 review Antiderivatives Areas and distances Riemann sums definite inte grals fundamental theorem of calculus total change Indefinite integrals net change theorem Techniques of integration 0 Substitution rule o Integration by parts o Trigonometric substitution trigonometric inte grals Partial fractions includes long division of poly nomiIas Rationalizing substitutions Approximate integration Midpoint Trapezoid and Simpson s rule Improper integrals includes I Hospital s rule Infi nite intervals discontinuous functions 7 Applications of integration Area between curves includes graphing func tions Volumes cross section and shell methods Work Average value of a function Arc length Moments and centers of mass 8 Differential equations modeling and solving 125 Final Exam Winter 2005 Problem 1 Evaluate the following integrals 2 2 0 51336 da 0 COSx o nsmxdx snx o 12005da da V5132 213 3 x44 d3 513 2132 Problem 2 o For b positive evaluate the integral b da 0 132 81 12 Simplify your answer o Evaluate the improper integral 00 da 0 132 81 12 7 or if it doesn t converge explain why Problem 3 Find the area of the region bounded by the curves y cos x y x2 1 1 O and 13 2 2 Problem 4 Let ft1t Vl I s4ds and let 933 21xftdt Find gquot2 Problem 5 Let b be a positive number and consider the region bounded by the curves y 5132 yx21 332 19 and 5131 1 Find the y coordinate of the center of mass of this region in terms of b 2 Because of the symmetry of this region the x coordinate of the center of mass is O For small values of b say b lt M the center of mass is in the region while for b gt M the center of mass is outside of the region Find M Problem 6 Consider the region bounded by yxSin7rx y 0 and 13 1 1 Set up an integral for the volume of the solid obtained by rotating this region about the y axis DO NOT EVALUATE 2 Set up an integral for the volume of the solid obtained by rotating this region about the x axis DO NOT EVALUATE Problem 7 A turkey has an internal tem perature of 60 degrees F At 1200 pm it is placed in an oven whose temperature is at 380 degrees 1 Newton s Law of Cooling states the rate of cooling or heating of an object is pro portional to the temperature difference be tween the object and its surroundings Use Newton s Law of Cooling to write down a differential equation and initial condition for the internal temperature y of the turkey at time 25 Take 75 O to be the time it is placed in the oven Your equation should involve an unknown constant k 2 After 2 hours it is observed that the tem perature of the turkey is 140 degrees Use this to find the constant k and the temper ature y as a function of time t 3 At what time will the temperature equal 170 degrees For this problem give a dec imal answer Problem 8 The region between the curve y f13 the line 1 1 and the line y 4 is rotated around the line 13 1 A formula for f13 is not known however we do have the following a table of values 1 115225 3 354 f13 o 3 7 12 19 28 4 Use Simpson s Rule to approximate the volume of this solid of revolution For this problem give a decimal answer June 1 2007 Announcements o This Week Finish 94 and Review for Final Exam 0 Final Exam Saturday June 2 2007 from 130 420pm in EEB 125 Instruction Page available on the web page Cumulative Exam Covers all material No graphing calculators yes scientific One 85 x 11 sheet of notes both sides May use any of the 20 integrals from the table on p 506 in 75 of the text without deriving them Show your work in evaluating any other integrals even if they are on your note sheet You must bring Picture ID to the Exam Do sample Finals Math 125 Materials webpage Today 0 Final Exam Review Winter 2005 TODiCS 120 review Antiderivatives Areas and distances Riemann sums definite inte grals fundamental theorem of calculus total change Indefinite integrals net change theorem Techniques of integration 0 Substitution rule o Integration by parts o Trigonometric substitution trigonometric inte grals Partial fractions includes long division of poly nomiIas Rationalizing substitutions Approximate integration Midpoint Trapezoid and Simpson s rule Improper integrals includes I Hospital s rule Infi nite intervals discontinuous functions 7 Applications of integration Area between curves includes graphing func tions Volumes cross section and shell methods Work Average value of a function Arc length Moments and centers of mass 8 Differential equations modeling and solving 125 Final Exam Winter 2005 Problem 1 Evaluate the following integrals 2 2 0 51336 da 0 COSx o nsmxdx snx o 12005da da V5132 213 3 x44 d3 513 2132 Problem 2 o For b positive evaluate the integral b da 0 132 81 12 Simplify your answer o Evaluate the improper integral 00 da 0 132 81 12 7 or if it doesn t converge explain why Problem 3 Find the area of the region bounded by the curves y cos x y x2 1 1 O and 13 2 2 Problem 4 Let ft1t Vl I s4ds and let 933 21xftdt Find gquot2 Problem 5 Let b be a positive number and consider the region bounded by the curves y 5132 yx21 332 19 and 5131 1 Find the y coordinate of the center of mass of this region in terms of b 2 Because of the symmetry of this region the x coordinate of the center of mass is O For small values of b say b lt M the center of mass is in the region while for b gt M the center of mass is outside of the region Find M Problem 6 Consider the region bounded by y x Sin7rx y O and 1 1 1 Set up an integral for the volume of the solid obtained by rotating this region about the y axis DO NOT EVALUATE 2 Set up an integral for the volume of the solid obtained by rotating this region about the x axis DO NOT EVALUATE Problem 7 A turkey has an internal tem perature of 60 degrees F At 1200 pm it is placed in an oven whose temperature is at 380 degrees 1 Newton s Law of Cooling states the rate of cooling or heating of an object is pro portional to the temperature difference be tween the object and its surroundings Use Newton s Law of Cooling to write down a differential equation and initial condition for the internal temperature y of the turkey at time 25 Take 75 O to be the time it is placed in the oven Your equation should involve an unknown constant k 2 After 2 hours it is observed that the tem perature of the turkey is 140 degrees Use this to find the constant k and the temper ature y as a function of time t 3 At what time will the temperature equal 170 degrees For this problem give a dec imal answer Problem 8 The region between the curve y f13 the line 1 1 and the line y 4 is rotated around the line 13 1 A formula for f13 is not known however we do have the following a table of values 1 115225 3 354 f13 o 3 7 12 19 28 4 Use Simpson s Rule to approximate the volume of this solid of revolution For this problem give a decimal answer May 14 2007 Announcements o This Week 81 Midterm Review and 83 o Homework 7 Week 7 problems Due Tuesday May 15 Hand in Thursday May 17 Covers 78 and 81 see web page for problems 0 Midterm 2 is this week Thursday May 17 Covers through 81 Start doing practice Midterms beware some cover ma terial up to 83 You may use any of the 20 integrals from the table on p 506 in 75 of the text without deriving them You must show your work in evaluating any other integrals even if they are on your note sheet There will be two consecutive review sessions on Wednes day afternoon both in MEB 234 Eina Ooka will begin at 230 and Mike Gaul will take over at 345 Eina and Mike will go over the following two midterms from Winter 2003 mine and Prof Bube s available on the Math 125 Material web page wwwmathwashingtonedu m125Quizzesweek8mid2k pdf wwwmathwashingtonedu m125Quizzesweek8mid2a pdf Today 0 Finish 81 Arc Length How do you compute the length of a curve given parametrically o Start 83 Applications to Physics and Engineering Moments and Centers of Mass The arc length function 5 of the curve Cxya x b yfx is the distance along the curve C from the ini tial point P0 afa to Q 13f13 ie sen x 1 W dt By the FTC d5 2 dy 2 1 f 22 i1 Thus 1 d82 dx2 dy2 Symbolically Lzds If the curve C is given parametrically by C atyt to S t S 21 then 1 becomes d5 2 da 2 dy 2 a a a and g xx t2 y t2 In this case t d t 2 L told dtzt01wxt23403 dt Problem 81 33 A hawk flying at 15ms at an altitude if 180m accidentally drops its prey The parabolic trajectory of the falling prey is described by the equation x2 180 y 45 until it hits the ground Here y is the height above the ground and 1 the horizontal distance traveled in meters Calculate the distance trav eled by the prey from the time it is dropped until the time it hits the ground Problem 81 34 A steady wind blows a kite due west The kite s height above the ground from horizontal position 1 O to 1 80 ft is given by 1 2 150 50 y 40x Find the distance traveled by the kite January 9 2008 Announcements o This Week 410 51 and 52 o You need to have the Math 125 course pack Always bring this to your Tu amp Th sections o Homework 1 Week 1 Problems Due Tuesday January 15 Covers 410 51 and 52 see web for assignment httpwwwmathwashingtonedu m125 Homeworksweeklbrobsbdf Today o Elementary table of particular antiderivatives again o 51 Areas and Distances Func on x n 75 1 COSx sin 10 sec2x 1 m2 l I m2 Par cuKN39an denva ve n1 n1 In sinx C0x tanx arcsnwx arcta n it Area Problem Find the area of the region S bounded by the graph ofa continuous function f where f13 2 O the x axis and the vertical lines 1 a and 5131 Sxy1a x b OSySfC BH Idea Divide S into n strips of the same width Abe a TL There is a corresponding division of a b into n subintervals x07x17 xlaxQL Ha mm lawn where 513020 xnzb xk1xkAaz for nggn l Compute Rn and Ln Rn fx1Ax fx2Ax fxnAx Ln fxoAx fx1Ax fxn1Ax Definition The area of the region S bounded by the graph of a non negative continuous func tion f the x axis and the vertical lines 1 a and 1 b is A Iim Iim L n gtooRn n gtoo n The velocity graph of a car accelerat ing from rest to a speed of 120 kmh over a period of 30 seconds is shown Estimate the distance traveled during this period 1 kmh 80 4O seconds March 28 2007 Announcements o This Week 410 51 and 52 o You need to have the Math 125 course pack Always bring this to your Tu amp Th sections o Homework 1 Week 1 Problems Due Tuesday April 3 Covers 410 51 and 52 see web for assignment httpwwwmathwashingtonedu m125 Homeworksweek1probspdf Today o Elementary table of particular antiderivatives o A velocityacceleration problem from 41O o 51 Areas and Distances Func on x n 75 1 COSx sin 10 sec2x 1 m2 l I m2 Par cuKN39an denva ve n1 n1 In sinx C0x tanx arcsnwx arctan LC Example A car braked with a constant deceleration of 16 fts2 producing skid marks measuring 200 ft before coming to a stop How fast was the car traveling when the brakes were first applied Area Problem Find the area of the region S bounded by the graph ofa continuous function f where f13 2 O the x axis and the vertical lines 1 a and 5131 Sxy1a x b OSySfC BH Idea Divide S into n strips of the same width Abe a TL There is a corresponding division of a b into n subintervals x07x17 xlaxQL Ha mm lawn where 513020 xnzb xk1xkAaz for nggn l Compute Rn and Ln Rn fx1Ax fx2Ax fxnAx Ln fxoAx fx1Ax fxn1Ax Definition The area of the region S bounded by the graph of a non negative continuous func tion f the x axis and the vertical lines 1 a and 1 b is A Iim Iim L n gtooRn n gtoo n

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