EXAM #2 MATH 231 CALCULUS 2- Santilli
Chapter 6,8 and 10- SHOW ALL WORK! Highlight answers for full credit.
1.) Sketch the graph of
sin53
32
r
in polar coordinates. Verify your graph by converting into rectangular
and then graphing again.
2.) Sketch the graph of
042 xyxy
.
3.) A hemispherical tank of radius 5 feet is filled with water. Find the work required to pump the water out through a
hole in the top of the tank.
4.) Use the slice method to determine the volume of the solid formed by revolving the region bounded by
2
2)( xxf
and
1)( xg
about the line
1y
. Check your results by then doing the problem with
cylindrical shells.
5.) Find the centroid of a lamina bounded by
2
4)( xxf
and
2)( xxg
. State both geometric moments,
x
G
and
y
G
.
6.) A swimming pool is 2 feet deep at one end and slopes down linearly to a depth of 10 feet. The pool is 40 feet
long and 30 feet wide with vertical sides. Find the hydrostatic fluid force against one of the 40 foot sides.
7.) Write the polar form of an ellipse with a focus at (4,0) and major vertices at (5,0) and (5,π). Solve for r as a
function of cos Ө.
8.) Given
x1
t1
and
yt
t1
, graph the corresponding rectangular equation and indicate the direction of
increasing t.
9.) You and a friend live 4 miles apart and are talking on the phone. You hear a clap of thunder from lightning in a
storm, and 18 seconds later your friend hears the thunder. (sound travels 1100 ft/sec)
a.) Find an equation that gives the possible places where the lightning could have occurred using the following
rectangular coordinate system: You are located at (-2,0) miles and your friend is located at (2,0) miles.
b.) Could the lightning have hit a baseball stadium that is 1 mile directly north of your home, i.e., located at (-
2,1)?
HINT: remember the definitions of the conic sections and determine which one applies to this problem.
10.) Find the length of the graph of
x
x
y2
1
6
3
over the interval
2 ,2/1x
.
11.) Find the surface area of the surface formed by revolving the graph
3
xy
on the interval
1 ,0x
about the
x-axis.
12.) Find the average velocity of a particle over the interval
x0,9
km that is traveling with a velocity profile
depicted by
v(x)
sin x 0x3 km
sinh x 3x7.2 k m
sinx 7. 2x9 km
m/s.
13.) Find the area bounded by
yx36x28x
and
yx24x
.