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Don't have my book or audio? You can download a free sample of my book and audio lectures containing Lessons 1 and 2: Did you read my tips on how to study and learn this course? If not, here is a link to those important suggestions: The department posted SOLUTIONS for Assignment 1 (these are not my solutions). Click here.The department posted SOLUTIONS for Assignment 1 (these are not my solutions). Click here.The department posted SOLUTIONS for Assignment 3 (these are not my solutions). Click here.Here is a link to the actual assignment, in case you don't have it: Study Lesson 12 (Improper Integrals), Lesson 13 (Arc Length and Surface Area), and the rest of Lesson 14 (Parametric Equations), and Lesson 15 (Polar Curves)
from my Calculus 2 book to prepare for this assignment.
Of course, always seek out assistance from my book, your course notes, etc. if you ever hit a question you don't understand, but try not to be learning things as you do an assignment. Learn first, then put your learning to the test.
Similar to my Lesson 12, #1, but these are not basic integrals. You will have to use the techniques you have learned to solve the integrals first. As always, go to the side and solve the indefinite integral first, then you can return to the definite
integral and proceed. As they have warned, these definite integrals are improper, so set up the appropriate limits to complete the problem.
You will have to use the Comparison Method as I outline in Lesson 12, #2 to solve these problems. Pay attention to their hint. You can only have one improper endpoint in an integral. If both endpoints are improper, you must split
into a sum of integrals.
Note that I use L to represent arc length (most profs use s, I chose to use L to distinguish it from S for surface area). I give you the formulas for dL, you then integrate dL to solve L.
Part (a) is very similar to the examples I
show in Lesson 13, #1.
Part (b) is using the parametric version of arc length as discussed in Lesson 14, using the dL formula I list. I only do one example, #5, part (d), part (i) where I use it to figure out the circumference of that parametric curve. In your problem, be careful to compute dx/dt and dy/dt correctly (there are product rules involved), and expect to see an opportunity to simplify the sum
of the squares through trig identities.
Use the polar arc length formula that I show you in Lesson 15, #3.
Back to Lesson 13. Now surface area using one of the common integrals I discuss in the arc length section.
Now we are using the parametric version of the arc length and surface area formulas. Sub the parametric formula for dL into the appropriate Surface Area formula you learned back in Lesson 13.
Be sure to graph this ellipse first to visualize it (plot the x-intercepts by subbing y=0 into the equation and plot the y-intercepts by subbing x=0 into the equation).
Now get ready to set up and solve the integral. I think you are best to
set this up as a dy integral, and focus on rotating the piece in the first quadrant. Note that creates exactly half of the surface, so you need only double that surface area to get the whole surface. This will be a challenging integral to solve.
Now we are back to using the parametric version of the arc length and surface area formulas. Sub the parametric formula for dL into the appropriate Surface Area formula you learned back in Lesson 13. Again, trig identities will come into
play.
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