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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 11:
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, for those of you who 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. 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.
Part (d)
Those of you who have my Intro Calculus book should look at my solutions to Lesson 11, Practise Problems 37 and 38. If you do not, be sure to look up odd functions and how that affects definite integrals.
Part (e)
There is a problem here! This is an improper integral
(Lesson 12). which I am sure you are not supposed to know how to handle here, because the function is undefined when y=0. Contact the prof about this. If you are given no direction or correction, I suggest you say it is an improper integral and don't do it. Perhaps, set up the indefinite integral and solve that, at least, but do not attempt to deal with the endpoints of that definite integral at this time.
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.
This question is NOT to be solved using elementary integral formulas! You couldn't solve it that way anyway.
First, split the problem up into two separate definite integrals using properties of antiderivatives. You can say it
is the definite integral from -5 to 0 of 2dx PLUS the definite integral from -5 to 0 of the square root expression. Don't forget to put dx at the end of each integral!
For the integral of 2, you have to draw a graph of f(x) = 2 from x=-5 to x=0, then find the area between the vertical lines x=-5 and x=0, between the graph of f(x) and the x-axis using geometric methods. You should notice the shape is just a rectangle, so
you can find the area by using the area of a rectangle.
For the other integral, draw the graph of f(x) = square root of (25- x^2) from x=-5 to x=0 and find the area between the curve and the x-axis. Hint: the graph is a quarter of a circle.
Don't forget that the original integral is the area of the first graph PLUS the area of the second graph.
Be sure to study Lesson 5 thoroughly before attempting this question. Make sure you sketch the graphs in this question to help visualize the region. Be careful! You will have to break the problem up into a couple of different regions to properly set up the
definite integral that computes the area of each region. What is the top and bottom curve for a specific region? Scan the graph carefully from left to right. When do the boundary curves change?
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 15, #2 and review polar graphing before attempting this question. Make sure you sketch the graph to help visualize the region. Take a look at my solutions to July 22, 1999 #8, December 12, 2000 #6, and
December 12, 2001 #8 for similar examples. You will need to use Half-Angle Identities to solve this integral as I illustrate in Lesson 7, #1(a) and (b).
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. Again, you will need to use Half-Angle Identities to solve this integral as I illustrate in Lesson 7, #1(a) and (b).
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. Remember, you must either
use two y formulas and do the integral of (y1-y2)dx or use two x formulas and use the integral of (x1-x2)dy. Don't you dare think you are subtracting those two trig equations! This is an ellipse. Exploit the symmetry. You can use either the x-axis or the y-axis, or both, as a boundary line.
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