## Third Midterm-Practice Test

You know the drill.

- Find the volume of the solid that lies under the plane and above the rectangle
- Find the volume of the solid enclosed by the paraboloid and the planes and
- For the integral , sketch the region of integration and change the order of integration.
- Find the volume of the solid enclosed by the parabolic cylinder and the planes
- Use a double integral to find the are of the region inside the cardioid and outside the circle
- Use polar coordinates to combine the sum below into one double integral, and evaluate it.
- A swimming pool is circular with a 40-ft diameter. The depth is constant along east-west lines and increases linearly from 2 ft at the south end to 7 ft at the north end. Find the volume of water in the pool.
- The boundary of a lamina consists of the semicircles and together with the portions of the –axis that join them. Find the center of mass of the lamina if the density at any point is proportional to its distance from the origin.
- Express the volume of the wedge in the first octant that is cut from the cylinder by the planes and as a triple integral.
- Evaluate the triple integral where is bounded by the cylinder and the planes and in the first octant.
- Identify the surface with equation in cylindrical coordinates given by
- Evaluate where is enclosed by the sphere in the first octant.
- Find the volume of the part of the ball that lies between the cones and
- Find the Jacobian of the transformation
- By using an appropriate change of variables, evaluate the integral where is the parallelogram enclosed by the lines and

### Leave a Reply Cancel reply

### We have moved!

On November 2014, I migrated this blog to blancosilva.github.io. Please update your bookmarks and RSS feeds accordingly.

### In the news:

### Recent Posts

- Migration
- Computational Geometry in Python
- Searching (again!?) for the SS Central America
- Jotto (5-letter Mastermind) in the NAO robot
- Robot stories
- Advanced Problem #18
- Book presentation at the USC Python Users Group
- Areas of Mathematics
- More on Lindenmayer Systems
- Some results related to the Feuerbach Point
- An Automatic Geometric Proof
- Sympy should suffice
- A nice application of Fatou’s Lemma
- Have a child, plant a tree, write a book
- Project Euler with Julia
- Seked
- Nezumi San
- Ruthless Thieves Stealing a Roll of Cloth
- Which one is the fake?
- Stones, balances, matrices
- Buy my book!
- Trigonometry
- Naïve Bayes
- Math still not the answer
- Sometimes Math is not the answer
- What if?
- Edge detection: The Convolution Approach
- OpArt
- So you want to be an Applied Mathematician
- Smallest Groups with Two Eyes
- The ultimate metapuzzle
- Where are the powers of two?
- Geolocation
- Boundary operators
- The Cantor Pairing Function
- El País’ weekly challenge
- Math Genealogy Project
- Basic Statistics in sage
- A Homework on the Web System
- Apollonian gaskets and circle inversion fractals
- Toying with basic fractals
- Unusual dice
- Wavelets in sage
- Edge detection: The Scale Space Theory
- Bertrand Paradox
- Voronoi mosaics
- Image Processing with numpy, scipy and matplotlibs in sage
- Super-Resolution Micrograph Reconstruction by Nonlocal-Means Applied to HAADF-STEM
- The Nonlocal-means Algorithm
- The hunt for a Bellman Function.
- Presentation: Hilbert Transform Pairs of Wavelets
- Presentation: The Dual-Tree Complex Wavelet Transform
- Presentation: Curvelets and Approximation Theory
- Poster: Curvelets vs. Wavelets (Mathematical Models of Natural Images)
- Wavelet Coefficients
- Modeling the Impact of Ebola and Bushmeat Hunting on Western Lowland Gorillas
- Triangulations
- Mechanical Geometry Theorem Proving

### Pages

- About me
- Books
- Curriculum Vitae
- Research
- Teaching
- Mathematical Imaging
- Introduction to the Theory of Distributions
- An Introduction to Algebraic Topology
- The Basic Practice of Statistics
- MA598R: Measure Theory
- MA122—Fall 2014
- MA141—Fall 2014
- MA142—Summer II 2012
- MA241—Spring 2014
- MA242—Fall 2013
- Past Sections
- MA122—Spring 2012
- MA122—Spring 2013
- Lesson Plan—section 007
- Lesson Plan—section 008
- Review for First part (section 007)
- Review for First part (section 008)
- Review for Second part (section 007)
- Review for Third part (section 007)
- Review for the Second part (section 008)
- Review for the Fourth part (section 007)
- Review for Third and Fourth parts (section 008)

- MA122—Fall 2013
- MA141—Spring 2010
- MA141—Fall 2012
- MA141—Spring 2013
- MA141—Fall 2013
- MA141—Spring 2014
- MA141—Summer 2014
- MA142—Fall 2011
- MA142—Spring 2012
- MA241—Fall 2011
- MA241—Fall 2012
- MA241—Spring 2013
- MA242—Fall 2012
- MA242—Spring 2012
- First Midterm Practice Test
- Second Midterm-Practice Test
- Third Midterm—Practice Test
- Review for the fourth part of the course
- Blake Rollins’ code in Java
- Ronen Rappaport’s project: messing with strings
- Sam Somani’s project: Understanding Black-Scholes
- Christina Papadimitriou’s project: Diffusion and Reaction in Catalysts

- Problem Solving
- Borsuk-Ulam and Fixed Point Theorems
- The Cantor Set
- The Jordan Curve Theorem
- My oldest plays the piano!
- How many hands did Ernie shake?
- A geometric fallacy
- What is the next number?
- Remainders
- Probability and Divisibility by 11
- Convex triangle-square polygons
- Thieves!
- Metapuzzles
- What day of the week?
- Exact Expression
- Chess puzzles
- Points on a plane
- Sequence of right triangles
- Sums of terms from Fibonacci
- Alleys
- Arithmetic Expressions
- Three circles
- Pick a point
- Bertrand Paradox
- Unusual dice
- El País’ weekly challenge
- Project Euler with Julia

- LaTeX

### Categories

aldebaran
algebra
algorithm
analysis
applied mathematics
approximation theory
books
calculus
catalyst
chemical engineering
circumcenter
coding
combinatorics
computational geometry
curvelets
data mining
Delaunay
denoising
differential geometry
edge detection
engineering
Euclidean geometry
fractal
functional analysis
geometry
harmonic analysis
history of math
image
image processing
imdb
Japan
L-system
LaTeX
latitude
Lindenmayer system
linear algebra
longitude
mathematical imaging
mathematics
matplotlibs
Measure Theory
metacritic
movie critics
movies
nonlocal means
numerical analysis
numpy
oxide
pattern recognition
physics
probability
programming
puzzles
python
sage
scanning transmission electron microscopy
scientific computing
scipy
segmentation
signal processing
statistics
stats
STEM
super-resolution
tex
tikz
tomography
topology
triangle
triangulation
trigonometry
Voronoi
voronoi diagram
wavelets
weights

### Archives

- November 2014
- September 2014
- August 2014
- July 2014
- June 2014
- March 2014
- December 2013
- October 2013
- September 2013
- July 2013
- June 2013
- April 2013
- January 2013
- December 2012
- August 2012
- July 2012
- June 2012
- May 2012
- April 2012
- November 2011
- September 2011
- August 2011
- June 2011
- May 2011
- April 2011
- February 2011
- January 2011
- December 2010
- May 2010
- April 2010
- September 2008
- September 2007
- August 2007

### @eseprimo

- .@PythonEggs @SciPyTip @Pybonacci import numpy as n, matplotlib.pyplot as p, scipy.special as s p.imshow(n.fromfunc… twitter.com/i/web/status/8… 1 week ago
- .@PythonEggs @SciPyTip @Pybonacci e.g. import scipy.linalg as s, matplotlib.pyplot as p p.imshow(s.lu(s.hadamard(12… twitter.com/i/web/status/8… 2 weeks ago
- In 140 characters (or less), can you generate (an approximation to) a Sierpinski gasket with #python? @PythonEggs @SciPyTip @Pybonacci share 2 weeks ago
- UofSC Mega Menger sponge is looking for a new home. Interested? @UofSC @USCResearch @ArtsSciencesUSC @MoMath1… twitter.com/i/web/status/8… 2 weeks ago
- RT @generativist: I'd support a gofundme for tight integration between VIM and @ProjectJupyter. Like, my vim, sitting in `~/.vim`. With ac… 3 weeks ago
- @fermatslibrary Boiler up! @PurdueScience 1 month ago
- Check out this #chess game: eseprimo vs randyorchangon123 - chess.com/livechess/game… 4 months ago
- eseprimo vs marouchkafish chess.com/live/game/1847… via @chesscom 4 months ago
- More than 1000 comments in the review test/forum I opened for my Applied Calculus. They mean business!… twitter.com/i/web/status/8… 4 months ago
- What makes a cult movie a cult movie? Example #1: The Commitments youtube.com/watch?v=gWnGnF… 4 months ago
- @scopatz No kidding. Nice job---very impressive. Thanks a lot! 4 months ago
- Oh whoa! #xonsh (a friggin shell language and command prompt BASED ON #python!!!) #GameChanger https://t.co/dqTVsiLQQC 4 months ago
- Check out this book: "Ocean of Storms" by Christopher Mari,… amzn.to/2fGRp4h https://t.co/XRFcsIqqs0 4 months ago
- @todoist who doesn't? 4 months ago
- Some of my students will be happy to know they are not the only ones struggling with the product/quotient rule...… twitter.com/i/web/status/8… 5 months ago
- @generativist amazon.com/gp/product/067… 5 months ago
- RT @pickover: Fractal based on Steiner chains. Source: bit.ly/2fuJGEg https://t.co/OZUUypUBeH 5 months ago
- Wasn't there a way to script @geogebra with #python ? I want my student to model his robot here, but the scripting… twitter.com/i/web/status/7… 5 months ago
- A must read (and henceforth a strong requirement) for all my students from now on. Brilliant insight, brutally hone… twitter.com/i/web/status/7… 5 months ago
- My wife was born to work for #buzzfeed , but somehow something happened in the way and ended up saving lives for a living #ILovethatWoman 5 months ago

### Math updates on arXiv.org

- Squeezed Fourier Meets Toeplitz Algebras. (arXiv:1704.05840v1 [quant-ph])
- A cohomological Seiberg-Witten invariant emerging from the adjunction inequality. (arXiv:1704.05859v1 [math.GT])
- A note on integrating products of linear forms over the unit simplex. (arXiv:1704.05867v1 [cs.PF])
- On Covering Monotonic Paths with Simple Random Walk. (arXiv:1704.05870v1 [math.PR])
- A matrix generalization of a theorem of Fine. (arXiv:1704.05872v1 [math.NT])
- Manton's five vortex equations from self-duality. (arXiv:1704.05875v1 [hep-th])
- Self-avoiding walks and connective constants. (arXiv:1704.05884v1 [math.CO])
- Waldschmidt constants for Stanley-Reisner ideals of a class of graphs. (arXiv:1704.05889v1 [math.AG])
- A Note on the Concentration of Spectral Measure of Wigner's Matrices. (arXiv:1704.05890v1 [math.PR])
- Unramified Godement-Jacquet theory for the spin similitude group. (arXiv:1704.05897v1 [math.NT])

### Computational Geometry updates on arXiv.org

- Proximal Nerve Complexes. A Computational Topology Approach. (arXiv:1704.05909v1 [cs.CG])
- Bounds on the number of discontinuities of Morton-type space-filling curves. (arXiv:1505.05055v4 [cs.CG] UPDATED)
- Bounding a global red-blue proportion using local conditions. (arXiv:1701.02200v3 [cs.CG] UPDATED)

### sagemath

- An error has occurred; the feed is probably down. Try again later.

For number 2, I want to see if I’ve set it up correctly (these ones confuse me somewhat).

I called u1 -> z = 1 and u2 -> z = 2 + x^2 + (y – 2)^2

So I set up the problem to be where you first take the integral of 1 dz from u1 to u2, and then integrate that resulting function over the domain D. Where D = {(x,y) : -1 <= x <= 1 ; 0 <= y <= 4}

Is that how you set up this problem?

yes

Wanna check my setup again.

For number 4 I got u1 -> z = 3y and u2 -> z = 2 + y

Then I set up the problem to be the integral of 1 dz from u1 to u2, and then integrate that resulting function over the domain D. Where D = {(x,y): 0 <= y <= 1; -sqrt(y) <= x <= sqrt(y)}

Tell me what you think. Thanks

That’s pretty good! Number 4 is a tough problem to visualize, yet you got the right equations. Would you care to explain how you obtained them to the rest of the class (here)?

Sure.

Well, as you said, it’s very hard to visualize, and that’s why I wanted to check it.

The first thing I noticed was that the equation is a parabolic cylinder which is just your classic at every point . So it runs parallel to the axis with its “spine” touching the axis.

Then you have the two planes and These are two flat planes that are parallel to the axis. You know this b/c is not in the equation. Now remember that just b/c they are parallel to the axis, doesn’t mean they are not tilted (b/c they are). They are tilted and they intersect. You can do a quick check of this and say that which means that they intersect at and

This is where it gets tricky to visualize. Here’s what we can say about the three surfaces…

This describes our enclosed solid area. Now we just have to figure out what the bounds of and are. Since the parabolic cylinder is parallel to the -axis it will NOT be our top or bottom surface ( and ). That means our two planes must be and (the boundaries). The domain of is pretty easy. Our parabolic cylinder is That means that goes from to (just solve for ). So what is the domain of Well the lowest will be is zero, But what about the highest it can be? Remember how the two planes intersect “inside” the mouth of the parabolic cylinder? That intersection is the max value of We found that to be 1 up above. So that gives us

Solving it is easy. We’re just finding the volume. So we take the integral of the function 1 with respect to and then integrate that resulting function on the domain and

Sorry for the long explanation. I tried to be as descriptive as possible since it’s so hard to visualize.

Shouldn’t the bounds of Y be: x^2 <= y <= 1? Since your cylinder's spine is on the Z axis.

Bobby’s description is good, but not the only one. Another possible description (using your bounds) for the region is as follows:

For number 6…

Does anyone have any clue where to start?

Number 10 is confusing me.

It doesn’t seem like there is an upper bound for the domain of z. Am I missing something?

There was a typo. Thanks for noticing! It is fixed now.

This one had me screwed up too. I am glad it was just a typo and nothing more complicated.

I noticed you did not assign homework from 15.7. Will there be questions from this section on the exam?

The problems are similar to those in 15.8. If you can do 15.8, then the ones in 15.7 should not be too much trouble. But yeah, there will be at least a question from that section.

I have been working on problem number 5 and I know what to do, however I did something wrong in the setup because I got a negative number. First, I set these equations equal, since both are equal to r, and found that there are two points of intersection and they are (pi/3) and (5pi/3). From there, I created 2 double integrals, 1 to account for the area between these points of the cardioid, and the other to subtract the area covered by the circle between these two points.

For the cardioid, I set up a double integral with the following bounds:

0 < r < 1+cos(theta)

(pi/3) < theta < 5(pi/3)

And integrated 1 to find the area of the region. I then set up the double integral for the circle using r going from 0 to 3cos(theta) and the same bounds for theta. Doing this gave me a negative number and I'm not exactly sure why.

You have to be careful how to compute the angles of intersection, that’s all.

To see why, using WolframAlpha, plot first between and . Note that this is exactly what we would expect. Now, plot between the same two angles: what do you observe?

In Calc II we learned how to treat this appropriately: See example 3 in section 10.4 (page 652)

Wouldn’t an easier approach be to just double the area that you get from doing the double integral between the two functions from pi/3 to pi?

Definitely! But be careful: for the cardioid it is indeed from to . For the circle, it is a different set of angles.