Math 22E: Additional assignment for Lab 3
Motivation/goal: Whenever you obtain an answer (in particular, with Mathematica), you need to make a rough estimate of your answer so as to provide a sanity check of whether your answer makes sense. This requiredaddendum to Lab 3reiterates the importance of the habit of checking your answer.
Each of the parts (e.g., (a-ADD), etc) below is worth half as many points as one part of the main Lab.
Note: Hand in you stapled1 andneatly presented2 handwritten work on these sheets. Attach additional sheets if necessary.Show all steps of your work whenever appropriate.
Make sure to explicitly answer each question posed below.
Exercise 1
(a-ADD)
Roughly approximate this region by a rectangle. The key word here is “roughly”: you should obtain the parameters of this rectangle by looking at the graph, not by doing any additional calculations to find the exact intercepts or the heightof the curve.
Sketch the rectangle and list its vertices:
(b-ADD)
Obtain a rough estimate of your answer in part (b). To this end, find the volume of the cylinder obtained by rotating the rectangle you found above about the y-axis. Recall that the volume V of a cylinder of radius r and height h is π r2h. Begin by sketching the cylinder and then list the corresponding values of r, h, and V.
1I will take off 5% of your grade if you hand in a pile of non-stapled sheets.
2I will reduce your grade by an amount left to my discretion in each particular case if your work is presented in a messy way and I have to waste time deciphering it.
Find the relative error (i.e., the error in percent) between your rough estimate and the exact answer in part (b) of the main Lab. Write the answer below.
Note: Since you are dealing with a rough estimate, it suffices (and, in fact, only makes sense) to round this error to an integer number of percent.
(c-ADD)
Obtain a rough estimate of your answer in part (c). To this end, find the volume of the cylinder obtained by rotating the rectangle you found above about the x-axis. List the corresponding values of r, h, and V:
Find the relative error (i.e., the error in percent) between your rough estimate and the exact answer in part (c) of the main Lab. Write the answer below rounded to an integer number of percent.
Exercise 2
(a-ADD)
In this case, the shape of the region is quite far from any rectangle or triangle. However, for the purpose of obtaining rough estimates to your answers, it may be sufficient to approximate the region by the rectangle with vertices at:
(1,2), (9,2), (9,5), (1,5).
Sketch this rectangle and find its area. Write the result below along with the relative error (with respect to the exact answer in part (a)) rounded to a number of percent that is a multiple of 5.
This, of course, is a rather poor estimate, but it still serves the purpose of providing a quick sanity check for your answers. Note that since this estimate is (significantly) greater than the exact answer, you should expect all your estimates below to also be greater than the corresponding exact answers.
(b-ADD)
Obtain a rough estimate of your first answer in part (b). To this end, find the volume of the cylinder obtained by rotating the rectangle you found above about the x-axis. List the corresponding values of rinner, router, h, and V:
Find the relative error (i.e., the error in percent) between your rough estimate and the exact value of the first answer in part (b) of the main Lab. Write the answer below rounded to a number of percent that is a multiple of 5.
Obtain a rough estimate of your second answer in part (b). To this end, find the volume of the hollow cylinder obtained by rotating the rectangle you found above about the line y=-5. List the corresponding values of rinner, router, h, and V:
Find the relative error (i.e., the error in percent) between your rough estimate and the exact value of the second answer in part (b) of the main Lab. Write the answer below rounded to a number of percent that is a multiple of 5.
(c-ADD)
Obtain a rough estimate of your first answer in part (c). To this end, find the volume of the cylinder obtained by rotating the rectangle you found above about the y-axis. List the corresponding values of rinner, router, h, and V:
Find the relative error (i.e., the error in percent) between your rough estimate and the exact value of the first answer in part (c) of the main Lab. Write the answer below rounded to a number of percent that is a multiple of 5. (Note that this estimate is the worst off all the estimates you have obtained in this Exercise. However, even this, rather poor, estimate is not too much worse than the estimate for the area obtained in part (a-ADD).)
Obtain a rough estimate of your second answer in part (c). To this end, find the volume of the hollow cylinder obtained by rotating the rectangle you found above about the line x=16. List the corresponding values of rinner, router, h, and V:
Finally, find the relative error (i.e., the error in percent) between your rough estimate and the exact value of the second answer in part (b) of the main Lab. Write the answer below rounded to a number of percent that is a multiple of 5.