ANT 2200 Troy University Cranium Theory of Evolution in Hominins Discussion Page 159:

Scholars are still working through the actual sequence of evolutionary changes in hominins that led to the development of bipedalism, large brains, social complexity, and tool-making abilities. Knowing what you now know about early hominins and how evolutionary processes work, can you theorize a possible sequence of evolutionary changes? (make sure you justify your answer)

I will send picture of the page in the book once the question is accepted. Should be at least one page Due April 24, 2020

1. Evaluate

!!

D

Math 281 – Quiz 6

Name__________________

Outside help permitted.

2

24xy dA where D is the region bounded by the graphs of y = x , and y – x = 2.

2. Reverse the order of integration and evaluate

3. Use polar coordinates to calculate

2

2

!!

D

9 3

!!

0

y

3 cos x 3 dx dy .

x dA where D is the region in the first quadrant that lies

between the circles x + y = 4 and x + y 2 = 25 .

3

”

4. Evaluate the integral ”

$ $

#0 #0

2

9% x 2

( x 2 + y 2 )3 2 dy dx by converting to polar coordinates.

Due April 24, 2020

Math 281 – Quiz 6

Outside help permitted.

1

5. Convert the integral ! !

0 0

coordinates and evaluate it.

1% x 2

!

x2 + y2

0

page 2 of 2

x 2 + y 2 dz dy dx from rectangular to cylindrical

6. Let E be the cone E = {( x, y, z ) : x 2 + y 2 & z & 4}.

a) Use cylindrical coordinates to find the volume, V, of E.

b) Find z =

1

V

!!!

E

z dV , the z-coordinate of the centroid of E.

7. Use spherical coordinates to find the volume of the solid between the sphere x 2 + y 2 + z 2 = 9 ,

and the cone z =

x 2 + y2

.

3

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