MATH 2080W15 - Assignment #5 | Complete Solution

MATH*2080W15 - Assignment #5


INSTRUCTIONS

1. Print this blank assignment right now. You'll be completing the following 4 pages, writing a summary of your solutions here, and submitting this completed assignment. No extra paper please - I won't accept it.

2. Note that for some questions, there is space provided for summary work. PLEASE DO NOT cram
every step into each answer space - but instead - transfer key steps from your rough work. For example, if you prepare a set of equations for solution using Partial Fractions, then show me how you get the equations. Then simply state that after solution the answers are etc. Or, when you perform integration, show me the start, the substitution, the completed set-up, and then say something like, after simpli cation. Then show me the nal step just before you integrate.

3. IF NOT ALREADY THERE, PLEASE MAKE A BOX AROUND YOUR FINAL ANSWER TO
MAKE IT EASIER TO LOCATE AND GRADE (thank you)

4. You are encouraged to work together. PLEASE - PLEASE - write-up your own solutions and submit your own individual write-up.

5. This assignment is worth 13% of your nal grade, graded in 1/2 % units, and is reduced
in size as a bonus to you.

6. This assignment is due Thursday April 2nd, by 4:30pm. I'll prepare a submission box
(orange box) in the library, where you will submit your assignments.

7. You are free to:
1) not do the assignment (your course grade is then computed out of 87% and scaled to 100%), or
2) do the assignment as usual.

Student #: Grade:
1. (11
2 mark) Create a double integral (with correct limits) that computes the volume bounded between
z = 􀀀x2 + 10 and z = y2 􀀀 10 where x 2 [-1,1] and y 2 [-1,1]. Solve it.
The double integral is:
The volume is:
2. (3 marks) Create a triple integral (with correct limits) that computes the volume bounded between
z = 􀀀x2 􀀀 y2 + 1 and the x 􀀀 y plane, in the positive orthont. Integrate in the order dz, dy, dx. Solve it,
showing key steps (the boxes) below.
The triple integral is:
After simpli cation, the double integral is:
After simpli cation, the single integral is:
The volume is:

c S.J. Gismondi (Instructor), 2015. All rights reserved.
3. (3 marks) Construct the triple integral, in the order dx, dy, dz that computes the nite volume in the
positive orthont bounded by x + 2y + 3z = 6. Complete the boxes below and compute the volume.
The triple integral is:
After simpli cation, the double integral is:
After simpli cation, the single integral is:
The volume is:
4. (3 marks) From above, construct the triple integral again but in the order dz, dy, dx and repeat the
volume computation. Complete the boxes below.
The triple integral is:
After simpli cation, the double integral is:
After simpli cation, the single integral is:
The volume is:

5. (21 2 marks) Consider the volume of the region bounded above by z = x2 + y2 + 1 and the x 􀀀 y plane, where x 2 [0,1] and y 2 [0,1]. Do the following.
a) Draw a picture of the volume such that the x axis is partitioned into four equal parts and the y axis
is partitioned into two equal parts. These will be called subregions in the x 􀀀 y plane. Be sure to clearly label x, y and z axes.
b) Explicitly construct/write the sum of the volumes of these eight rectangular boxes, each box having a base de ned by a subregion, and where z
i is the height of each box on each subregion (Compute z i in the very middle of each subregion.). This is call the Riemann sum. Show your work here.
c) Compute the numerical value of the Riemann sum. Show your work here.