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Don't have my book or audio lectures? You can download a sample containing some of these lessons here: Did you read my tips on how to study and learn Math
1700? If not, here is a link to those important suggestions: These are tips for the assignments in the Distance/Online Math 1700 course, but I strongly recommend that you do this assignment as homework even if you are taking the classroom lecture section of the course. These assignments are very good (and challenging) practice. It
is possible that you are doing the topics in a different order in the classroom lecture sections, so you may need to wait until later before tackling this assignment.
Here is a link to the actual assignment, in case you don't have it:
You need to study Lesson 4 (The Method of u Substitution), Lesson 5 (Area Between 2 Curves), and Lesson 6 (Volumes), from my
Calculus 2 book to prepare for this assignment! I think you should find this assignment fairly straightforward if you do thoroughly study and do all the Practise Problems I give you in these lessons.
These are classic u substitution integrals (actually, the first one doesn't require u sub) as taught in Lesson 4 of my book. Be careful, in part (c) you can't use u for your symbol since u is already in the problem. Do a t substitution or
an x substitution or something.
These are again similar to the questions I do in Lesson 4, just definite integrals this time,
I recommend you first set up and solve the indefinite integral, then use that solution to compute the answer to the definite integral. There
is a technique where you can change the endpoints of a definite integral if you have done a u substitution, but I find that unhelpful. Frequently in this course, you will have to use complex methods to solve an integral. Carrying the endpoints of a definite integral just adds an unnecessary complication and can lead many students to lose track or express their solutions erroneously.
A far more prudent approach, is, when given a definite integral,
first set up the associated indefinite integral and solve it first, then return to the definite integral and complete your solution.
Part (b) Note that the derivative of inverse secant comes in handy here. I don't mention the derivative in Lesson 1 of my book because the derivative is too picky. It can be stated several ways depending on the domain of you are using and also depending on the range of inverse secant.
There is no universal answer, so it is not something I think would come up on an exam. The link below is the derivative most commonly used, but it is only valid if x>1. But that is relevant here, since the definite integral is from 1 to 2.
This question is NOT to be solved using elementary integral formulas! You couldn't solve it that way anyway until you have studied Lesson 9 (Integrating by Trig Substitution), and you aren't allowed to do it that way even if you have studied that lesson.
First, split the problem into two different definite integrals. The definite integral of 2 from -3 to 3, plus the definite integral of the square root part from -3 to 3.
The first integral:
You have to draw a graph of f(x) = 2 (a horizontal line) from x=-3 to x=3, then find the area between the vertical lines x=-3 and x=3, below the graph of f(x) and above the x-axis
using geometric methods. You should notice the shape is a simple rectangle.
The second integral:
Draw the graph of f(x) = square root of (9 - x^2) from x=-3 to x=3 and find the area between the curve and the x-axis. Hint: the graph is a semicircle, so you know the area of this shape.
Then, of course, simply add those two areas together.
This is a typical area problem as taught in Lesson 5 of my book. Be careful! You definitely must sketch the graph of these three curves and visualize the area they describe. You will need to draw a vertical line to cut the region up
into two separate parts in order to identify the "top" curve and the "bottom" curve for each part.
Put another way, you can only find the area between two curves, so which two curves are relevant for each part?
Typical Volume questions as taught in Lesson 6 of my book. Make sure you look at my question 3 especially for examples rotating around lines other than the x- or y-axis.
This is just a case of applying the Average Value formula as stated in Lesson 2 of my book (page 7). There is an example of me applying this formula in the old exam from March 1997 (page 192, solution on page 235). Note that, since t is in minutes, this is
a=0 to a=60.
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