Math 1320 Lab2
Instructor: Bridget Fan
Name:
Due: 01/28/16 in class
Show all work including necessary graphs to get full credits!
1. Calculate the length of the curve defined parametrically by y
from I = 0 to t= 2ii-.
U
0
J9 J
=
et
cos(t), x
P
=
c sin(t)
ecu 4e
-
{.
=
(f)f
(-)
eCoi )
r)r
zJo
(-): e2
s e’k
-$e
:
2
[(cJ
e rcoij
i
1
I
[ci42s
ze-if •2
Math 1320
Lab2
-
Page 2 of 6
Name:
2. A steady wind blows a kite due west. The height above the ground from horizontal
position r 0 is given by
11
y=(+l)2 _(x+1)2.
(a) Find the position where the kite falls onto the ground.
y
kW
e
(x* : 0
uW.s onw
tQ
I
(N )
-
4
(ct1)
‘fI I /
--i
)hi
3
(b) Find the distance traveled by the kite before it fell on the ground. (Hint: 1 +
is a perfect square.)
:
flRjck
-
()2
I
ii=2
1
23RX
P
(4\)
G9
\Q
-
_cJ_
-
o1
—
4
:(
-
—
2
‘J%)I
L)3
I
—
I
I
Lab2
IViath 1320
-
Name:
Page 3 of 6
3. Consider a thin wire of length 3 cm, positioned so that it lies in the region 0 <
given by:
The wire then has a density in mg cm -1•
x
< 3.
71
(a) Draw the graph of the density function. Find the mass of the wire in mg. (Hint:the
mass can be viewed as the area of the region bounded by the density function.)
j
‘3
2 6A(&’)
x
1
(b) Use the symmetry of .)(.r) to guess where
-ie..
ne
&sfj
is
the center of the mass,
(Pf’r
ts
.
f
ks
(c) Write down an integral expressing
with your guess from part (b).
and evaluate the integral to check if it matches
f ?c’)(c)tx 4f
thc
jJc +)d
z
k6n)]
f2X(-
-frf2cox
0
3
;4?S)
SoSfl ()a
i COs ) o
:*3.-(-)-Q
-
—
—7
—
*$s(ck
L
I 4
—
—
3
-
-L-(’]
iViath 1320
Lab2
-
Name:
Page 4 of 6
4. Recall that there are three methods to estimate definite integral: Midpoint Rule, Trape
zoid Rule and Simpson’s Rule. Assume that we want to evaluate the integral of function
1(x) on interval [cij, b]. First, we will divide the interval into n. subintervals with length
°.
=
Then, we will estimate the area under the curve on each subinterval
as
following:
x
—
=
A,
f
0
K
0
K
3
K
9
=
f(x)dx
+...
Ax[f(i)+f(
)
9
1
K
Midepeint Rule
il
=
) + f(x]Ar
1
[f(x_
f (x)clx
=
SO
K,
2
K
1 +2
(A
A +... + A)
[f(.ro) + 2f(xi) +
. . .
+
f(x)j
13
Trepeecid Rule
A
=
)
_
1
[f(x
)]x
1
+ 4f(xj) + f(x
+A
1
(A
+
ff’ 3
=
Ku
Xi
13
.03
Simpson’s Rule
X
3
X
...
[f(xo)+4f(x1)+2f(x2)+.
.
Math 1320
Lab2
-
Name:
Page 5 of 6
14.
(a) Use r
=
and all three different methods to estimate the following integral:
+
f2
cos(1 + x)d
2
0
=
s\e Cpccto(
[co.s
()
(°- ()± (°
fr
f
& [-(o)+ 2-(()
C
(i)
f2(O
(4J
i-2{(’) +2f)4.f(2)]
j- .2 (Q (2)f2
Cos()+s (3))
:o-9q L129
TcosU>Jx
:
Lf(o)((±)42t( ‘f()
-[(os(’)+
o6ols5
(b) Calculate the integral and compare the result with part (a), which method gives
the closest estimation?
(Os(ik
=
(3) -Sin (i)
= -o1oooq16t
t(’1
(—)+os()
Math 1320
Lab2
-
Name:
Page 6 of 6
5. Suppose that you are gathering water from a well by pulling a rope attached to a bucket
of water. As the force of
F acts on the bucket, this requires a work 14 given by
gravity
T4
=
/
FG(y)dy
Jo
where d = 8 rn is the depth of the well and y = 0 corresponds to the bottom of the
well. If the mass of the rope is negligible, the force of gravity on the bucket is given by
Fo(y) = rn(y)g, where g = 9.81 in s
2 and in(y) is the mass of the bucket.
(a) Suppose that the mass of the bucket is constant, m(y)
=
5 kg. Find W.
b9.Jj
= 5.g,\
392t\
(b) Now suppose that the bucket has the same initial mass rn(0) = 5 kg, but is contin
uously leaking so that its mass is reduced by a constant rate of 0.25 kg for every
meter it is raised. Find m(y) and W.
ij);5- O
w1r
3
-
(o -si)