Double Integral Examples

Here it is... the first post where we(Spring 2011 AMAT 214 class) all collaborate to provide examples for the current course material. You can sign in to blogger with the username: albanymath@yahoo.com, and the password I send in your email(and announced in class) see the last post for how I would prefer you to add your examples. I'll go quickly over in class (and maybe make a video?) showing you how to do it more in depth.

Here are the first examples.


1. Find the volume of the solid under the plane $x+y+z=3$ and above the region in the first quadrant bounded by $y=x^2$ and $y=x^4$. solution

2. Find the volume for the region bounded by the cone $z=\sqrt{2x^2+2y^2}$ and the sphere $$x^2+y^2+z^2=12$$.

Solution:

3. Evaluate by converting the integral to polar coordinates.
$$\int_{-2}^{2}\int_{-\sqrt{4-y^2}}^{\sqrt{4-y^2}}xy^2~dA$$
Solution:

4. Find the area of the region bounded by the ellipse $16x^2+25y^2=100$ by using
$$A(D)=\iint_{D}dA,$$ with the transformation $x=\frac{u}{4}$ and $y=\frac{v}{5}$. Solution.

5. Evaluate the integral
$$\int_{R}\sin\left(\frac{y-x}{y+x}\right)dA,$$
where $R$ is the triangular region with vertices $(0,0)$, $(1,0)$, and $(0,1)$.
Solution: We'll use the substitution $u=y-x$ and $v=y+x$. This transforms $R$ into a triangular region in the $u-v$ plane with vertices $(0,0)$, $(1,1)$, and $(-1,1)$, which can be described in terms of $u$ and $v$ as follows:
$$-v\leq u\leq v$$
$$-1\leq v\leq 1$$
so we have:
$$\int_{R}\sin\left(\frac{y-x}{y+x}\right)dA=\int_0^1\int_{-v}^{v}\sin\left(\frac{u}{v}\right)\frac{\partial(x,y)}{\partial(u,v)}dudv$$
$$=\int_0^1\int_{-v}^{v}\sin\left(\frac{u}{v}\right)\frac{1}{2}dudv=\frac{1}{2}\int_0^1 -v\cos\left(\frac{u}{v}\right)_{-v}^{v}~dv$$
$$=\int_0^1 \frac{-v}{2}(\cos(1)-\cos(-1))dv=0$$


There you have it. I'll let these go for a week or so and then I will post solutions. In the coming weeks I may scale the webassign back a little to give you time to do this. One last thing, maybe you should "sign up" for the examples in the comments section so no two people are working on the same question.

Change of Variables and a new idea

Before I get started with the main focus of today's post I would like to share a few links I've found that might prove helpful in understanding change of variables in multiple integrals. Here is one, and here is another. Here is a video:


An unexpectedly popular portion of the last post was the ability for you to submit solutions to some of the examples. Due to this popularity, I am going to give the class to opportunity to post solutions here at least once a week. I have emailed the entire class a username and password that will allow you to edit posts in order to post your solution. Indicate on your solution your name so I can give you credit. The first solution will fulfill the "participation" portion of your grade. Each additional solution will earn you some extra points. The following is a list in order of preference of how to post solutions:

1. Use LaTex to type up your solution, here is an online complier.  Then upload the pdf to google docs using the username and password I provided. Once in google docs you can get a link for the file and attach that to the example on the blog.
2. Use anything else you type up your solution and attach it to the example.
3. Handwrite your solution and take a picture, upload the picture, and link it to the example.

Here is an example of what I mean. Check out the source code here.

In the next day or so I'll make a whole post of current examples and you can post your solutions as you wish.

Techniques of Integration

Now that we are embarking into the world of multiple integrals it is essential that everyone has a good handle on basic techniques for solving single integrals.

Basic Integrals:
These are problems where the integrand is a "basic" function like $x^n$, $e^x$,$\frac{1}{1+x^2}$,  $\sin(x)$, or $\cos(x)$. There are several(1,2,3) places on the web to find these basic indefinite integrals. While I see no problem in using charts like these as a reference, I recommend not becoming too dependent on them.

u-substitution:
I like to separate these types of integrals into two types. The first is "baby u-substitution" these are integrals where the substitution can be done "in your head" without writing anything else down. Examples are as follows(click to see a solution):



The second type of u-substitution are slightly more complicated. Some examples are as follows(click to see a solution):



Integration by parts
There are two main type of integrals where the method of integration by parts is used. The first involve integrands which are the product of a polynomial and a tracendental function (like $e^x$ or $\sin x$). An example is as follows:

The second type are integrals involving inverse functions, like the following(click to see a solution).


Here is a pretty good write up on nearly everything about integration by parts.

I'll add to this page a bit later, but I want to go ahead and post it so you guys can see some examples of tricky integrals.

More Written Homework Hints

Before I start with the hints I would like to remind you that in the original sheet I handed out there is a typo in question 2. I have fixed it in the downloads section. Also, recall that you can write up a solution to the bonus question from exam one as bonus on this assignment.  On to the hints.

1. Your best bet is to write $\mathbf{a}=\left<a_{1},a_{2},a_{3}\right>$ and grind it out. There is a clever way around this if you are interested.
2. Its hard to give a hint to this without giving it away. 
3. see earlier post.
4. Begin by building the position function for the projectile as we did in class. Once this is done you want to solve for the angle at two points.  The first of which corresponds to the rock just barely making inside the city wall, $(100,15)$. The second corresponds to the rock landing within the city while nearly going to far. I'll let you find the coordinates for this second point. During the calculation you should end up with a quadratic equation in $\tan \alpha$, where $\alpha$ is the angle. So brush up on trig identities if you get stuck.
5. This problem uses the chain rule which we only covered recently in class.  It is best to start with a two variable version of the problem and then generalize. Suppose $f(x)$ is homogeneous of degree m. Thus, 
. 

Now let $u=tx$ and $v=ty$ so we have

.
Now take the derivative of each side of the equation with respect to $t$.  The rest follows pretty easily. I'll let you figure out the equation you want to end up with(its the 2 variable version of the one on the homework).  After working out this simplified version the version on the assignment shouldn't be so bad.

If anyone has any other hints post them in the comments.  If someone wants to post the final answer to question 4(the values of the angles) feel free, then everyone can check their work.

Exam 1 Corrections and hint for written homework 3.

You have a chance to correct problems from the first exam. Here is the important information:

• Corrections will be due this thursday afternoon(March 3).  If I'm not around slip it under my door. It will be considered "on time" if its under my door when I arrive on Friday morning.
• Each question was worth 12 points. You can earn up to half of the points you missed for each question.
• I don't want your test book back, but make sure to write how many points you earned for each problem on the sheet you turn in.
• If you didn't keep the exam, you can find it in the download page of this blog.
• The bonus question will become a bonus question in first written homework assignment.

Now for a hint on number three of the written homework.  Recall, this problem involves calculating the curvature of a curve with parametric equations:

, and .

First of all I would write r(t)=x(t)i+y(t)j, and then choose one of the curvature equations to work with. It will be important to use the fundamental theorem of calculus:
 .