Mth 231 written assignment 6. MODULE 12 WRITTEN ASSGINEMNT 6 ONLY QUESTIONS #1-20PLEASE TYPE ANSWER ON WORD DOCUMENT PROVIDED IN MATH LAB OR MATH TYPE. LEAVE SPACE AFTER ANSWER FOR COMMENTS.PLEASE SHOW ALL WORK FOR ANSWER.I HAVE ATTACHED TWO COPIES.WORD DOCUMENT IS TO BE USED TO TYPE ANSWERS UNDER EACH QUESTION AND PDF IS A REFERENCE WITH ANSWERS ATTACHED.
Microsoft Word - MTH231- written assignment 5_ Claiborne working Copyright © 2021 by Thomas Edison State University. All rights reserved. Back to Top What height allows you to have the largest volume? 16. Draw the given optimization problem and solve: Find the dimensions of the closed cylinder volume V = 16π that has the least amount of surface area. 17. 18. Evaluate the limit 19. Evaluate the limit 20. Evaluate the limit Module 10—Written Assignment 5 1. Let Ln denote the left-endpoint sum using n subintervals and let Rn denote the corresponding right-endpoint sum. Compute the indicated left and right sums for the given functions on the indicated interval. R4 for g(x) = cos(πx) on [0, 1] =1/4[cos(π1/4)+cos(π2/4)+cos(π3/4)+cos(π4/4)] =1/4[cos(π/4)+cos(π/2)+cos(3π/4)+cos(π)] R4=1/4[0.707+0-0.707-1] = -0.25 Copyright © 2021 by Thomas Edison State University. All rights reserved. Back to Top 2. Express the following endpoint sum in sigma notation but do not evaluate them. F(2/5k)=√4-(2/5k)2 = √4-4/25k2 = 1/5√100-4k2 3. Express the limit as integral: 4. Evaluate the integral using area formula: =1/2πr2 = 2π 5. Show that the average value of sin2t over [0, 2π] is equal to 1/2 Without further calculation, determine whether the average value of sin2t over [0, π] is also equal to 1/2. 6. Use the Fundamental Theorem of Calculus, Part 1, to find the derivative: =F’(x)=ecosx Copyright © 2021 by Thomas Edison State University. All rights reserved. Back to Top 7. Use the Fundamental Theorem of Calculus, Part 1, to find the derivative: =F’(x)=cos2x 8. Evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2. 9. Evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2. F’(4)=-2(1/4)-2(4)1/2 =-1/2+1 F’(1)=-2(1/1)+2(1)1/4 =-2+2 -1/2 +1 +0 = 1/2 10. Evaluate 11. Suppose that a particle moves along a straight line with acceleration defined by a(t) = t − 3, where 0 ≤ t ≤ 6 (in meters per second). Find the velocity and displacement at time t and the total distance traveled up to t = 6 if v(0) = 3 and d(0) = 0. 12. A ball is thrown upward from a height of 3 m at an initial speed of 60 m/sec. Acceleration resulting from gravity is −9.8 m/sec2. Neglecting air resistance, solve for the velocity v(t) and the height h(t) of the ball t seconds after it is thrown and before it returns to the ground. Copyright © 2021 by Thomas Edison State University. All rights reserved. Back to Top 13. Find the antiderivative using the indicated substitution 14. Use a suitable change of variables to determine the indefinite integral. 15. Use a suitable change of variables to determine the indefinite integral. 16. Use a suitable change of variables to determine the indefinite integral. 17. The area of a semicircle of radius 1 can be expressed as Use the substitution x = cost to express the area of a semicircle as the integral of a trigonometric function. You do not need to compute the integral. 18. Find each indefinite integral by using appropriate substitution. 19. Find each indefinite integral by using appropriate substitution. 20. Verify by differentiation that ∫lnx dx = x(ln x − 1) + C, then use appropriate changes of variables to compute the integral. Copyright © 2021 by Thomas Edison State University. All rights reserved. Back to Top Module 12—Written Assignment 6 1. Determine the area of the region between the two curves by integrating over the y-axis. 2. Graph the equations and shade the area of the region between the curves. Determine its area by integrating over the x-axis. 3. Graph the equations and shade the area of the region between the curves. Determine its area by integrating over the x-axis. 4. Graph the equations and shade the area of the region between the curves. If necessary, break the region into subregions to determine its entire area. 5. Graph the equations and shade the area of the region between the curves. Determine its area by integrating over the x-axis or y-axis, whichever seems more convenient. Copyright © 2021 by Thomas Edison State University. All rights reserved. Back to Top 6. Find the exact area of the region bounded by the given equations if possible. If you are unable to determine the intersection points analytically, use a calculator to approximate the intersection points with three decimal places and determine the approximate area of the region. 7. The largest triangle with a base on the x-axis that fits inside the upper half of the unit circle y2 + x2 = 1 is given by y = 1 + x and y = 1 − x. See the following figure. What is the area inside the semicircle but outside the triangle? 8. Use shells to find the volumes of the given solids. Note that the rotated regions lie between the curve and the x-axis and are rotated around the y-axis. 9. Use shells to find the volume generated by rotating the regions between the given curve and y = 0 around the x-axis. 10. Find the length of the functions over the given interval. 11. Find the length of the functions over the given interval. If you cannot, use technology to get the answer. Copyright © 2021 by Thomas Edison State University. All rights reserved. Back to Top 12. Find the lengths of the function of y over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it. 13. Find the mass of the one dimensional object: A car antenna that is 3 ft long (starting at x = 0) and has a density function of ρ(x) = 3x + 2 lb/ft 14. Find the mass of the one dimensional object: A pencil that is 4 in. long (starting at x = 2) and has a density function of ρ(x) = 5/x oz/in. 15. A spring has a natural length of 10 cm. It takes 2 J to stretch the spring to 15 cm. How much work would it take to stretch the spring from 15 cm to 20 cm? 16. A force of F = 20x − x3 N stretches a nonlinear spring by x meters. What work is required to stretch the spring from x = 0 to x = 2 m? 17. Find the work done when you push a box along the floor 2 m, when you apply a constant force of F = 100 N. 18. What is the work done moving a particle from x = 0 to x = 1 m if the force acting on it is F = 3x2 N? 19. A cylinder of depth H and cross-sectional area A stands full of water at density ρ. Compute the work to pump all the water to the top. 20. For the cylinder in the preceding exercise (problem 19), compute the work to pump all the water to the top if the cylinder is only half full. Module 12—Written Assignment 6 1. Determine the area of the region between the two curves by integrating over the y-axis. =18-(-18) = 36 square units over the y-axis 2. Graph the equations and shade the area of the region between the curves. Determine its area by integrating over the x-axis. (e-1)-1/2[(e2-1)/e] square units over the x-axis 3. Graph the equations and shade the area of the region between the curves. Determine its area by integrating over the x-axis. This study source was downloaded by 100000794010510 from CourseHero.com on 08-03-2021 21:46:05 GMT -05:00 https://www.coursehero.com/file/81179810/6pdf/ Th is stu dy re so ur ce w as sh are d v ia Co ur se He ro .co m https://www.coursehero.com/file/81179810/6pdf/ The area of the region, R between the two given curves is 2 square units. 4. Graph the equations and shade the area of the region between the curves. If necessary, break the region into subregions to determine its entire area. The area of the region, R between the two given curves is 4.875 square units 5. Graph the equations and shade the area of the region between the curves. Determine its area by integrating over the x-axis or y-axis, whichever seems more convenient. The area between curves 2y=x and y+y3=x is 0.5 sq. units 6. Find the exact area of the region bounded by the given equations if possible. If you are unable to determine the intersection points analytically, use a calculator to approximate the intersection points with three decimal