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SusanAthena
SusanAthena, Master's Degree
Category: Calculus and Above
Satisfied Customers: 102
Experience:  Tutor for Algebra, Geometry, Statistics. Explaining math in plain English.
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Calculus III questions

Customer Question

Calculus III questions
Submitted: 18 days ago.
Category: Calculus and Above
Customer: replied 18 days ago.
questions are attached
Customer: replied 18 days ago.
Needs to be submitted anytime on Wednesday NOV, 16.
Expert:  Mr. Glenn replied 18 days ago.

Hi

May I help you?

Customer: replied 18 days ago.
please look attached.
Expert:  Mr. Glenn replied 18 days ago.

ok, will update you later

Expert:  Mr. Glenn replied 17 days ago.

hi

I have not realized these problems are really tough, due to the nature of the bounded area.

For the centroid,
.
xbar = [2*(2.58492) + 2.739105283(7.029682312)] / [2*(2.92847) + 7.029682312 ] = 1.8953671

for ybar it is at the axis of symmetry which is the x axis, so ybar is zero.

so centroid is (xbar, ybar) = (1.8953671, 0)

(interestingly, the centroid is outside the area, and this is common for unusual shapes like this (crescent) horshoes, etc.)
.
(By the way, to make it simpler to compute for the centroid, I divided the crescent using the line x = 2, so the bounded region was divided by three subparts)
.

I am sorry I will need more time to answer for the remaining problems, as this will require more analysis.

Customer: replied 17 days ago.
there are really tough and take time to solve. I technically need them in 2 hours and half but I also need some time to write them down before the submission. Please try to finish them asap
Expert:  Mr. Glenn replied 17 days ago.

will try

Customer: replied 17 days ago.
Are you still gonna post solutions?
Expert:  Mr. Glenn replied 17 days ago.

Hi, I think I need more time, (maybe at least a day) to answer the remaining question. (right now it is early dawn in our place).
.
For the first part (centroid)
.
solution is (when partitioned by x = 2)
.
area of subpart 1 (the upper and the lower part has the same area)
.
integral (sqrt(8 -(x-1)^2) - sqrt (4 - x^2) dx from -1.5 to 2 = 2.92847 (area)
.
integral x(sqrt(8 -(x-1)^2) - sqrt (4 - x^2) dx from -1.5 to 2 = 2.58492
.

For the middle part:
.
Area = ​integral 2*sqrt (8 - (x-1)^2)dx from 2 to (1+sqrt8) = 7.029682312
.
and

.
​integral 2x*sqrt (8 - (x-1)^2)dx from 2 to (1+sqrt8) = 19.3765217636757
.
.
So
.
Centroid of the Crescent region:
.
xbar = [2*(2.58492) + 19.3765217636757] / [2*(2.92847) + 7.029682312 ]
.
= 1.90479407
.
for ybar it is at the axis of symmetry which is the x axis, so ybar is zero.
.
so centroid of the crescent shape is (xbar, ybar) = (1.90479407, 0)

Customer: replied 17 days ago.
Day is too much. I need it sooner like in 12 hours max.
Expert:  Mr. Glenn replied 17 days ago.

will try again, part b and c I think is possible, I am still figuring out for the last part ( with reference to z) will update you later (still early dawn here)

Expert:  Mr. Glenn replied 17 days ago.

Really, this crescent figure makes it more difficult to compute the moment of inertia (not so common shape)

Customer: replied 17 days ago.
It's really stressing me out that my solutions are never submitted on time. If you can't do it in 12 hours pleases let me know now. Other than that I will probably request a refund and I don't want to waste your time or mine.
Expert:  Mr. Glenn replied 17 days ago.

ok

Customer: replied 17 days ago.
Are you gonna submit them on time?
Expert:  Mr. Glenn replied 17 days ago.

will try ( I will update you later)