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Question: 0 0 and r 1 is shown below the region...

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0, 0, and r -1 is shown below: The region R is bounded by y1+r, Question 1: Using the Disk Method, set up and evaluate an integral that gives the volume of the solid obtained when R is revolved about the r-axis. Give both an eract answer and the approximate answer to 8 decimal places. The eract answer (in terms of π) is: The approrimate answer to 8 decimal places is:
To demonstrate the procedure for approximating the volume, we treat the case that appears in the images above; we use 4 rectangles of equal width and require that the height of each rectangle, and hence the radius of each disk, is determined by the value of the function y - f(x) evaluated at the righthand endpoint of each rectangle. 1-0 1 Notice that the width Δι-- For the dark shaded second rectangle: The righthand endpoint is located at 12-2Ar =-. . The height of the rectangle, and hence radius of the second dark shaded disk, is: . The volume of a disk of radius R and thickness h is given by: Thus, the volume of the second dark disk is: 64 Using this procedure, fill in the table below. You should include one sample calculation in the box provided on the next page; it is not necessary to include all of the calculations! Make sure to calculate the eract volume of each disk in terms of π! 25a 2 4 64
Use this space to show how you obtained your values for n, Rn and V in the table for one value of n (other than n 2). The approximate volume of the region is the sum of the volumes of the four disks. Question 2: What is the approximate volume of the solid obtained above? Repot both the eract result of Vi + ½ + ½ + va in terms of π, and the decimal expansion of this answer to 8 decimal places The exact result of Vi + + Vs+ Va (in terms of π) is: The approzimate answer to 8 decimal places is: Is the answer close to the actual volume of the solid you computed at the beginning of the problem? How could we obtain a volume closer to the exact volume of the solid? The answer, as usual, is to use more rectangles! Of course, it would be a pain to do this by hand! Indeed, if we use 100 slices, we would have to find the volumes of 100 disks in a manner similar to the above and add them all together. For a computer though, this task is simple!
Question 3: By using the indicated number of slices, calculate the approximate volume of the solid to 8 decimal places and record your results in the table below. The result for n- 4 is recorded, and the result for n 100 is given as well. The file in the Projects folder gives explicit instructions to set up the worksheet for n 4, and to check that you understand the procedure, make sure that your answer for n -100 matches the one given here. 7.17289416 10 50 100 5.91163962 500 1000 You should have computed that to 8 decimal places, the eract volume is 5.86430628 cubic units. Do the numbers in your table get closer to this as n increases? In this approximation step, we could find a formula that gives the approximate volume of the sold in terms of n. To do this, we would need to compute the volune AVi of the We then would have to add the volumes of all of these together. Letting V denote the actual volume of the solid, we could write:
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