Practice Problems for April 27 2011

1. Evaluate the surface integral

$$\iint_S z~dS,$$

where $S$ is part of the paraboloid $z=x^2+y^2$ under the plane $z=9$.

Solution:
Using the tools from class we can write this integral as follows.
$$\iint_S z~dS=\iint_D(x^2+y^2)\sqrt{1+4x^2+4y^2}dA$$
Where $D$ is the disk $x^2+y^2\leq9$. After changing to polar coordinates we have
$$\int_0^{2\pi}\int_0^3r^3\sqrt{1+4r^2}drd\theta=2\pi\int_0^3r^3\sqrt{1+4r^2}dr,$$
which may be evaluated using trigonometric substitution. Let $r=\frac{1}{2}\tan\phi$, so $dr=\frac{1}{2}\sec^2\phi$ and $\sqrt{1+4r^2}=\sqrt{\sec^2\phi}=\sec\phi$. Thus the integral is transformed into
$$\int\frac{1}{16}\tan^3\phi\sec^3\phi d\phi=\frac{1}{16}\int \tan\phi(\sec^2\phi-1)\sec^3\phi d\phi$$
$$=\frac{1}{16}\int (\sec^5\phi-\sec^3\phi)\tan\phi\sec\phi d\phi$$
Now use $u$-substitution with $u=\sec\phi$ and thus $du=\tan\phi\sec\phi d\phi$ and so the integral becomes
$$\frac{1}{16}\int(u^5-u^3)du=\frac{1}{96}u^6-\frac{1}{64}u^4$$
reversing the substitutions we have
$$=\frac{1}{96}\sec^6\phi-\frac{1}{64}\sec^4\phi$$.
Then using $\sec\phi=\sqrt{1+\tan^2\phi}$ this becomes
$$=\frac{1}{96}(1+4r^2)^3-\frac{1}{64}(1+4r^2)^2.$$
Now evaluate at $r=3$ and $r=0$ and we are all done.

2. Evaluate the surface integral

$$\iint_S x~dS,$$

where $S$ is the bottom half of the sphere $x^2+y^2+z^2=4$.

Posted by: Angel Castro

Solution

3. Evaluate the surface integral

$$\iint_S (x^2+y^2+z^2)dS,$$

where $S$ is the cylinder $x^2+z^2=1$ between $y=-1$ and $y=1$, together with the top and bottom disks.

Solution: by Dima Tarasevich

4. Suppose $\mathbf{F}(x,y,z)=x\mathbf{i}-z\mathbf{j}+y\mathbf{k}$ and $S$ is the part of the sphere $x^2+y^2+z^2=16$ in the first octant. Find

$$\iint_S \mathbf{F}\cdot d\mathbf{S}.$$

5. Suppose $\mathbf{F}(x,y,z)=xy\mathbf{i}+yz\mathbf{k}$ and $S$ is parameterized by $\mathbf{r}(u,v)=(u+v)\mathbf{i}+uv\mathbf{j}+(u-v)\mathbf{k}$ for $(u,v)\in [0,1]\times[0,1]$. Find

$$\iint_S \mathbf{F}\cdot d\mathbf{S}.$$

SOLUTION:

$\mathbf{r}u X \mathbf{r}v = (-v-u)\mathbf{i}+ 2\mathbf{j} + (u-v)\mathbf{k}$

$\mathbf{F}\cdot \mathbf{r}u X \mathbf{r}v = (u+v)(-v-u) +2uv + (u-v)(u-v)$

$$\int_0^1\int_0^1(u+v)(-v-u)\mathbf{i} +2uv\mathbf{j} + (u-v)(u-v)\mathbf{k}$$

$$\int_0^1\int_0^1 2uv dudv = 1/2$$

Charlie

Practice Problems for April 26 2011

Here is a collection of problems related to what we did today in class. Feel free to add solutions

1. Find the parametric equations of the cylinder $\frac{y^2}{4}+9z^2=1$

2. Find the parametric equations of the sphere $x^2+y^2+z^2=25$ above the cone $z=\sqrt{x^2+y^2}$. What values of the parameters are needed?

3. Find the surface area of the part of the sphere $x^2+y^2+z^2=1$ between $z=-\frac{1}{2}$ and $z=\frac{1}{2}$.

Solution: First we'll parameterize the sphere by $x=\cos\theta\sin\phi$, $y=\sin\theta\sin\phi$, and $z=\cos\phi$. So we can write $\mathbf{r}(\theta,\phi)=\cos\theta\sin\phi\mathbf{i}+\sin\theta\sin\phi\mathbf{j}+\cos\phi\mathbf{k}$. In order to find the values of $\theta$ and $\phi$ that "draw" the needed portion of the sphere we set  $z=\pm \frac{1}{2}$ and so

$$x^2+y^2=\frac{3}{4}$$

after plugging in the parametric equations we are left with

$$\sin^2\phi=\frac{3}{4}$$

and thus

$$\sin\phi=\pm\frac{\sqrt{3}}{2}.$$

This corresponds to $\frac{\pi}{3}\leq\phi\leq\frac{5\pi}{3}$(looking it up here). The values of $\theta$ are from $0$ to $2\pi$. Now we are ready.  Recall the formula for surface area

$$A(S)=\iint_S dS=\iint_D |\mathbf{r}_u\times\mathbf{r}_v|dA,$$

where $D$ are the necessary values of $u$ and $v$ to create $S$. First we'll calculate

$$\mathbf{r}_{\phi}\times\mathbf{r}_\theta=\mbox{Det}\pmatrix{\mathbf{i} & \mathbf{j} & \mathbf{k} \\ \cos\theta\cos\phi & \sin\theta\cos\phi & -\sin\phi \\ -\sin\theta\sin\phi & \cos\theta\sin\phi & 0}.$$
Putting it all together
$$A(S)=\int_{0}^{2\pi}\int_{\pi/3}^{5\pi/3}\sin\phi d\phi d\theta$$
$$~~2\pi(\cos(\pi/3)-\cos(5\pi/3))=2\pi$$.

4. Find the area of the boundary of the region enclosed by $z=x^2+y^2$ and $z=9$.

5. Find the area of the surface $z=\sqrt{x^3}+\sqrt{y^3}$ with $(x,y)\in[0,1]\times [0,1]$.

Solution: Recall in this special case we can use
$$A(S)=\int_D \sqrt{1+(\frac{\partial z}{\partial x})^2(\frac{\partial z}{\partial y})^2}dA$$, where
$D=[0,1]\times [0,1]$, $\frac{\partial z}{\partial x}=\frac{3}{2}\sqrt{x}$, and $\frac{\partial z}{\partial y}=\frac{3}{2}\sqrt{y}$. So
$$A(S)=\int_0^1\int_0^1\sqrt{1+\frac{9}{4}x+\frac{9}{4}y}dxdy$$
$$~~~=\int_0^1\frac{8}{27}((\frac{13}{4}+\frac{9}{4}y)^{3/2}-(1+\frac{9}{4}y)^{3/2})dy,$$
and you can take it from here.

6. Find the are of the part of the cylinder $x^2+y^2=4$ between $z=0$ and $z=x+1$.

Exam 2 Corrections and New Examples

As we did last time, I am allowing you to make corrections on the second test. The rules are the same as they were before:

• Corrections will be due this thursday afternoon(April 14). 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 second written homework assignment.

Now for a few examples:

Example 1: Let $\mathbf{F}(x,y)=(2x+ye^{xy})\mathbf{i}+(2y+xe^{xy})\mathbf{j}$ and $C$ be the curve parameterized by $x=te^{t^2-t}+1$,
$y=\sin \frac{\pi}{2}t+\cos \frac{\pi}{2}t$ for $t\in[0,1]$. Find
$$\int_C \mathbf{F}\cdot d\mathbf{r}$$.

Example 2: Let $C$ be the boundary of the triangle with vertices $(0,0)$, $(1,1)$, and $(1,0)$. Find the following integral.

$$\oint_{C} xydx+2x^2y^2 dy$$

-Abigail Walker


Example 3: Let $C$ be the ellipse defined by $\frac{x^2}{4}+\frac{y^2}{9}=1$. Evaluate

$$\oint_C ydx+xydy.$$

Example 4: Evaluate

$$\iiint_{E}\sqrt{x^2+y^2}dV,$$

where $E$ is the region enclosed by the planes $z=0$, $z=x+y+5$, and the
cylinders $x^2+y^2=4$ and $x^2+y^2=9$.


Example 5: Let $\mathbf{F}(x,y,z)=(xyz+2z^2)\mathbf{i}+\cos(xyz)\mathbf{j}+xy^2z^3\mathbf{k}$. Find $\mbox{div}~\mathbf{F}$ and $\mbox{curl}~\mathbf{F}$.

SOLUTION:

-Sabrina Campfield

Example 6: Let $\mathbf{F}(x,y,z)=e^{x^2+y^2+z^2}\mathbf{i}-\arctan(xyz)\mathbf{k}$. Find $\mbox{div}~\mathbf{F}$ and $\mbox{curl}~\mathbf{F}$.

Example 7: If the components of $\mathbf{F}$ have continuous second partial derivatives, verify the identity.
$$\mbox{div}~\mbox{curl}~\mathbf{F}=0$$

Solution:

Damian Miraglia


Example 8: Verify the identity.
$$\mbox{div}(\mathbf{F}\times\mathbf{G})=\mathbf{G}\cdot\mbox{curl}(\mathbf{F})-\mathbf{F}\cdot\mbox{curl}(\mathbf{G})$$

Written Homework 2 Hints

1.  First off, if you look at this problem the right way it turns becomes very similar to one we worked out in class and was on the second test. Here is a similar problem.

$$\int_0^1e^{\mbox{max}\{x,1-x\}}dx$$.

First we'll split up the region of integration into two pieces, one where $x$ is larger and one where $1-x$ is larger. Observe that $1-x$ is larger on the interval $[0,1/2]$ and $x$ is larger on the interval $[1/2,1]$. So we can write

$$\int_0^1e^{\mbox{max}\{x,1-x\}}dx=\int_{0}^{1/2}e^{1-x}dx+\int_{1/2}^1 e^xdx,$$

which are fairly simple.

2.  The hint written on the assignment is sufficient.

3.  Use the formula for 4 dimensional volume(similar to the one for area).

$$V(D)=\iiiint_{D}dV,$$ 

and use change of variables to write 

$$dV=\left|\frac{\partial(x,y,z,w)}{\partial(r,\theta,s,\varphi)}\right|,$$

and use the technique here to take the determinant. Be careful about the bounds of integration for $r$ and $s$.

Thats all for now, we'll have the technology to solve 4 and 5 after this week.

Exam 2 Notes and Examples

Our second exam is coming up day after tomorrow. In light of this I would like to relay a few things that I have said in class.

1. You can present a question from the review sheet in class tomorrow(Wednesday 6 April) to earn 5 bonus points on the exam.
2. You will be allowed a formula sheet that you write. It may be up to one page(only front), and may include no examples.
3. I'm putting some examples you should look at below, feel free to add your solution to fulfill your "online collaboration" credit.

Example 1:

Calculate the following line integral, where $C$ is the left half of a circle with radius 3.

$$ \int_{C}(x^{2}y+xy^2)ds $$

Solution: Taking inspiration from polar coordinates, we will parameterize $C$ by $\mathbf{r}(t)=3\cos t\mathbf{i}+3\sin t\mathbf{j}$, with $t\in[\pi/2,3\pi/2]$. A simple calculation gives

$$ds=\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}=3.$$

So the integral becomes

$$\int_{C}(x^{2}y+xy^2)ds=81\int_{\pi/2}^{3\pi/2}(\cos^2t\sin t+\cos t\sin^2t)dt$$

$$=(27\sin^3t)_{\pi/2}^{3\pi/2}=-54.$$

Example 2:

Calculate the line integral, where $C$ is the line segment from $(1,-2)$ to $(4,1)$.

$$\int_{C}(xe^y+y\sin(x))ds$$
Solution:
We can parameterize the line segment using $\mathbf{r}(t)=\left<1,-2\right>(1-t)+\left<4,1\right>t$, so we have $x=1+3t$ and $y=-2+3t$. This gives us $ds=\sqrt{9+9}dt=3\sqrt{2}dt$ and thus
$$\int_{C}(xe^y+y\sin(x))ds=3\sqrt{2}\int_0^1((1+3t)e^{-2+3t}+(-2+3t)\sin(1+3t))dt$$

$$~~~~=[te^{-2+3t}-\frac{1}{3}(2+3t)\cos(1+3t)+\frac{1}{3}\sin(1+3t)]_{t=0}^{t=1}$$
$$~~~~=\mbox{you can get the rest}$$

Example 3:

Let $C$ be the curve made up of the line segment from $(-1,0)$ to $(0,0)$, the curve $y=x^2$ from $(0,0)$ to $(1,1)$, and the line segment from $(1,1)$ to $(1,2)$. Calculate the line integral.

$$\int_{C}xds.$$

Solution

I would suggest working these out before the exam and then if you want to post a solution do so between thursday and tuesday. This way you still have time to study. If there are no solutions by next tuesday I'll post solutions, but I warn you guys there are only going to be a limited number of chances to write these solutions so you might consider "getting it done" soon.

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:
 .

Exam 1 Wrap Up and Bonus Question Hint

For the exam tomorrow you can expect 8 calculations. Don't expect any basic calculations involving vectors. Most of that material will be covered by calculations in other problems(i.e. cross product by finding B and , length of a vector in the arclength integral). There will be one abstract problem(like 17 from the review) involving some vector identity.

As far as the bonus, it could be one of three things:
1. A four dimensional generalization of the cross product.
2. A projectile motion question similar to the "baseball in the wind" example from class.
3. Combining curves to form something "nice" in terms of curvature.

Good Luck!

Projectile Motion

A classic application of working with position, velocity, and acceleration in terms of vector function is to answer questions pertaining to projectile motion. Like what we did in class today(2/10/2011). It turns out there are a ton of helpful resources on the web for these types of problems.

Here is an hour long video from an MIT introductory physics course. He doesn't start from the begining of the topic, but does give several good examples that follow from what we did in class.

Watch it on Academic Earth

Here is another example:



If you guys find anything else that is helpful post it in the comments.

Intersections of Planes Example

A question in class today pointed out some minor detail that I left out in finding the equation of a line which is the intersection of two planes. Lets look at an example where the method in class does not work, see how to fix it, and then look at what went wrong.

Lets consider the planes $2x+y+z=2$ and $4x+2y-3z=4$. If we set $z=0$, we are left with the two equations $2x+y=2$ and $4x+2y=4$, which have infinitely many solutions. We can't continue.

A good way around this is simply to set $x=0$ or $y=0$ and solve the way we did in class. If we set $x=0$, we are given the equations $y+z=2$ and $2y-3z=4$. These have a solution of $y=2$ and $z=0$, giving us the point $(0,2,0)$. Now we are all set. We can cross product the normal vectors from each plane to get a vector pointing in the direction of the line and finish by writing the equation.

What went wrong? Geometrically, setting z=0 is equivalent to looking for the intersection of the three planes $2x+y+z=2$, $4x+2y-3z=4$, and $z=0$. In this case the line of intersection of $2x+y+z=2$ and $4x+2y-3z=4$ is contained in the plane $z=0$, so we gain no information by setting $z=0$. In the picture below $2x+y+z=2$ is the red plane, $4x+2y-3z=4$ is the blue plane, and $z=0$ is the yellow plane. Notice how the line of intersection of the red and blue planes is contained in the yellow plane.

We fixed this by setting $x=0$. Which is illustrated in the following graph(the red and blue are as above and $x=0$ is yellow.

While the intersection of the planes is not in the frame it is clear that they will intersect in a point, giving us a point which is on our line.

3D visualization

Before I get going with the topic of today's post I would like to add another resource I found for understanding the geometry involved in the dot product and cross product. Just follow the links to play.

While it is possible to solve homework questions without having a good handle on the geometry of the functions we are dealing with, I wouldn't advise it.  Here are some 3 dimensional function graphing applets that you can use.  For lines in 3 dimensions it is best to use the parametric equations, making this a good choice:

For graphing planes the following works well(make sure and use a * for multiplication, so 3x=3*x):

First Real Post: Book-keeping and Vectors.

Hopefully everyone received the email with the syllabus. In case you didn't if put it in the "Downloads" section of this blog.  I recommend that everyone in the course "follow" this blog as I will be posting helpful information including possible extra-credit opportunities.

On to the course material.  We have so far covered vectors up to the start of the cross product.  Here is a good video covering the dot product from MIT Open-Courseware.

Here is a video covering the cross-product:




Comment to this post with more online resources for the course material up to this point. This will count toward your participation grade. I'll give you until class on tuesday to comment, after which I'll post some resources I've found.