Follow:
|
There was a typo in my tips for question 8 that I have fixed below. It was Lesson 14, #4 I wanted you to look at, not #2.
Try a Free Sample of Grant's Audio Lectures (on sale for only $20)
|
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.Here is a link to the actual assignment, in case you don't have it: Study Lesson 2 (The Fundamental Theorem of Calculus), Lesson 4 (The Method of u Substitution, Lesson 5 (Area Between 2 Curves), Lesson 6 (Volumes), Lesson 7 (Integrals
of Trigonometric Functions), Lesson 14 (Parametric Equations: ONLY THE AREA PART, my Lesson 14, #4, #5(d-part ii), and Lesson 15 (Polar Curves: ONLY THE AREA PART, my Lesson 15, #2) from my Calculus 2 book to prepare for this assignment.
Note that you can safely IGNORE my Lesson 3 (Riemann Sums). They would not make you do that stuff on exams. Just know the definition, and,
it is possible that they could make you apply the definition similar to what I do in my Lesson 3, #4.
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.
Technically, Lesson 1 is not taught until later in the course, but I STRONGLY
RECOMMEND that you study this lesson NOW. It is a good review of trigonometry and derivatives which will stand you in good stead for the rest of the course. Also, inverse trig shows up in all of the lessons in this course, so it is best to know it right away and then you can study the lessons thoroughly, including applications where inverse trig shows up. In class, they often skip the examples that require inverse trig for now because they haven't taught it yet.
BUT, when they do finally teach inverse trig, they never go back and show you the examples in the earlier applications that required inverse trig, leaving a gap in your practice that can come back to haunt you on exams. These are classic derivative applications of the Fundamental Theorem of Calculus. Study my Lesson 2, #2 thoroughly to prepare.
Be sure to study Lesson 2, Lesson 4, and Lesson 7 thoroughly before attempting this question. You are not properly prepared to solve these integrals until you have completed all three of those lessons. Be sure to memorize the Elementary
Integrals on page 1 of my book, and remember my advice about index cards on page 6 of my book while you study these lessons. Make the cards as you are practicing each integral for the first time and you will not have the tedious task of making over a hundred cards all at once later on.
Suggestion: Never carry the endpoints of a definite integral while solving it if it requires a more advanced technique like u
substitution. It just clutters the problem up, and, if you retain the endpoints while changing the variable to u, YOU ARE WRONG, because those endpoints are x values, not u values. You either must change the endpoints to match your new variable, or clearly label them as x values. All of which is just a nuisance.
Instead, go to the side of your paper and set up the related indefinite integral and solve that instead. Once you have
solved the indefinite integral, return to the original definite integral and apply your solution to that. I do this by splitting the page into two columns. The original definite integral is in the left column, and I put the indefinite integral in the right column. That way, when I have solved the indefinite integral, I have all this space in the left column to remind me that I am not finished yet! There is still the definite integral to
complete.
Be sure to study Lesson 2, Lesson 4, and Lesson 7 thoroughly before attempting this question. You are not properly prepared to solve these integrals until you have completed all three of those lessons. Be sure to memorize the Elementary
Integrals on page 1 of my book, and remember my advice about index cards on page 6 of my book while you study these lessons. Make the cards as you are practicing each integral for the first time and you will not have the tedious task of making over a hundred cards all at once later on.
Be sure to study Lesson 5 thoroughly before attempting this question. Make sure you sketch the graphs in both of these questions to help visualize the region.
For part (b), find the points that make y=0, and
then note that y is both the positive and negative square root of the right side of that equation. You can find the area between the positive square root and the negative square root, or you can exploit the symmetry of the graph, if you prefer.
Be sure to study Lesson 6 thoroughly before attempting this question. Make sure you sketch the graph to help visualize the region.
Note that I show you how to set up integrals when rotating around unusual axes in my
Lesson 6, #3, but you must study the whole lesson to properly prepare for this question.
Be sure to study Lesson 6 thoroughly before attempting this question. Make sure you sketch the graph to help visualize the region.
Note that I show you how to set up integrals when rotating around unusual axes in my Lesson
6, #3, but you must study the whole lesson to properly prepare for this question.
Be sure to study Lesson 6 thoroughly before attempting this question. Make sure you sketch the graph to help visualize the region.
Note that I show you how to set up integrals when rotating around unusual axes in my Lesson
6, #3, but you must study the whole lesson to properly prepare for this question.
Be sure to study Lesson 14, #4 and review parametric graphing before attempting this question. You can also take a look at my #5 (d), part (iii). Make sure you sketch the graph to help visualize the region.
Be sure to study Lesson 15, #2 and review polar graphing before attempting this question. Make sure you sketch the graphs to help visualize the region.
BE CAREFUL! To properly visualize the boundaries of the area, start at the
initial angle and rotate counterclockwise, sweeping the region that you have shaded inside both curves. What angle is it where the curves intersect? The angle lines give you the diagonal lines of a pizza slice, and the endpoints of your definite integral. What curve marks the outer edge of your pizza slice? Is there an inner curve cutting off the pointy part of the pizza slice near the origin, like someone took a bite out of the pizza? If there is, that tells you to
subtract the inner curve area from the outer curve area to get the area between the curves. If not, then there is no area to subtract.
HINT: You will need to add up to separate integrals to find the complete area described.
Be sure to study Lesson 15, #2 and review polar graphing before attempting this question. Make sure you sketch the graph to help visualize the region. Here, be very aware of the sections where 1-2cosθ is negative, since the absolute value has made r
positive there. Since the area formula squares r, the absolute value won't affect your integral, since squaring the absolute value is akin to squaring the function without absolute value signs. But, the absolute value certainly affects the graph, and so the region described.
|
|
