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Quiz 2 - Math 141 (6380) Instructions: • The quiz is worth 100 points. There are 10 problems, each worth 10 points. Your score on the quiz will be converted to a percentage and posted in your assignment folder with comments. • This quiz is open book and open notes, and you may take as long as you like on it provided that you submit the quiz no later than the due date posted in our course schedule of the syllabus. You may refer to your textbook, notes, and online classroom materials, but you may not consult anyone. • You must show all of your work to receive full credit. If a problem does not seem to require work, write a sentence or two to justify your answer. • Please write neatly. Illegible answers will be assumed to be incorrect. • Please remember to show ALL of your work on every problem. Read the basic rules for showing work below BEFORE you start working on the quiz: 1. Each step should show the complete expression or equation rather than a piece of it. 2. Each new step should follow logically from the previous step, following rules of algebra. 3. Each new step should be beneath the previous step. 4. The equal sign, =, should only connect equal numbers or expressions. • This quiz is due at 11:59 PM (Eastern Time) on Tuesday, November 8. *********************** At the end of your quiz you must include the following dated statement with your name typed in lieu of a signature. Without this signed statement you will receive a zero. I have completed this quiz myself, working independently and not consulting anyone except the instructor. I have neither given nor received help on this quiz. Name: Date: 1 Quiz 2 - Volume, Arc Length, Surface Area 1. Find the volume of the solid obtained by rotating about the x-axis the region under the curve y = √ x from 0 to 1. Sketch the solid and explain how you form the integral that calculates the volume of the solid. 2. Find the volume of a pyramid whose base is a square with side ` and whose height is h. Sketch the solid and indicate how you form the integral. 3. Find the volume of the solid obtained by rotating the region bounded by y = x3, y = x, and x ≥ 0 about the x-axis. Sketch the bounded region which will be rotated about the x-axis. 4. Find the exact length of the curve y = 1 4 x2 − 1 2 lnx, where 1 ≤ x ≤ 2. 5. Find the exact length of the curve y = x3 3 + 1 4x , where 1 ≤ x ≤ 2. (See chapter 5-2 for reference) 6. Find the exact length of the curve y = ln(sec x), where 0 ≤ x ≤ π 4 . (See chapter 5-2 for reference) 7. Find the area of the surface obtained by rotating the curve y = √ 1 + 4x, where 1 ≤ x ≤ 5, about the x-axis. (Use chapter 5-2) 8. Chapter 5-3, Problem 32. 9. Chapter 5-3, Problem 34. 10. Chapter 5-4, Problem 14. 2 definite integral. Partition the time interval [a,b] into short subintervals. For the interval [ty 4]: force = f(c;) for any c; in [ti_y. tj] distance moved = work = fej) \| (Axi/Ay Toulwok = 3 work slong cach subimenat} = 3 16) var (ay? & =H J 0 \ an? + @y/a0® dt = total work along the path (x(0. 300) ta In problems 31 - 35, find the total work along the given parametric path. If necessary. approximate the value of the integral using your calculator. f is in pounds, x and y are in feet, i tes. 316) = 1. x0) = cos. y(0 =sin().0< 12%.="" 32.="" {0="1." xo="ty0" =="" 0="r=1." ban="r." x0="r" yn=""><><1 34.="" £0)="sin(n." x(0)="20,y(1)" =31,0="" r="x." (fig.27)="" 35.="" 1()="1." x()="cos(n)" y(t)="sin)," 0s="" 12x="" (fig.="" 28).="" (can="" you="" find="" a="" geometric="" way="" to="" calculate="" the="" shaded="" area?)="" mporary="" calculus="" in="" problems="" 11-26,="" sketch="" the="" region="" bounded="" between="" the="" given="" functions="" on="" the="" interval="" and="" calculate="" the="" centroid="" of="" each="" region="" (use="" simpson's="" rule="" with="" n="20" if="" necessary).="" plot="" the="" location="" of="" the="" centroid="" on="" your="" skeich="" of="" the="" region="" .y="x" and="" the="" x-axis="" for="">
<3. .="" y="x?" and="" the="" x-axis="" for="" 2x2.="" y="x"> and the line y=4 for 2
<2. y="sin(x)" and="" the="" x-axis="" for="" 0.x=""><7. y="4-x"> and the x-axis for 25x52 x2 and y=x for 0sx<1 y="9-x" and="" y="3" for="" 0x3,="" y="y=VR" andthe="" x-axis="" for="" 0x9.="" 20.="" y="In(x)" and="" the="" x-axis="" for="" 1="">
<1 22. y=x% and the 2x for 0x22 an empty one foot square tin box (fig. 25) weighs 10 pounds and its center of mass is 6 inches above the bottom of the box. when the box is full with 60 pounds of liquid. the center of mass of the box-liquid system is again 6 inches of mass of the box-liquid d in the box. (b) what height of liquid in the bottom of the box results in the box-liqui system having the lowest center of mass (and the greatest stability)? 22.="" y="x%" and="" the="" 2x="" for="" 0x22="" an="" empty="" one="" foot="" square="" tin="" box="" (fig.="" 25)="" weighs="" 10="" pounds="" and="" its="" center="" of="" mass="" is="" 6="" inches="" above="" the="" bottom="" of="" the="" box.="" when="" the="" box="" is="" full="" with="" 60="" pounds="" of="" liquid.="" the="" center="" of="" mass="" of="" the="" box-liquid="" system="" is="" again="" 6="" inches="" of="" mass="" of="" the="" box-liquid="" d="" in="" the="" box.="" (b)="" what="" height="" of="" liquid="" in="" the="" bottom="" of="" the="" box="" results="" in="" the="" box-liqui="" system="" having="" the="" lowest="" center="" of="" mass="" (and="" the="" greatest="">