N
MATH132O: hitegration Techniques and Area Between
Curves
Iustnictor: Laura Strube
26 August 2015
Name:
Student ID:
(A
1. Integration by Parts
Integrate:
eç
f’
2
r
dr using the integration by parts method.
2
4
V
1nd V
V
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A
LÀ
(-u-o
r
Ow=2rch’
r
2r
-
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-
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-
S2
u &ubsf#H
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=
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1
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.:
3. Area under a Curve
. Begin by sketching the
2
and g() = 2x r
Find the area between the curves f(x) =
2 to determine the limits of integration.
curves and solving x’ = 2x x
—
—
()(): 2
ZX
-)(
+
((Z_
-
Zx
_(_2x+ —‘
bonur cir’
rL (YIZ (A) b-’-
ck
0
or
W
- Cx)
2.X-
Zx —X
2
x
I
:
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j)
(x-*--2’)
=0
Ix
( -4-1--
)(
‘1.-
iX2
-
x’
-4-
—
0
Dcnn+
-+X+2.
2
X
‘t-eQ
i
rz,cDi3
4. Integrating with Respect to y
2 = 4 and the line 4x 3y 4 by integrating with
Find the area between the parabola y
2 3y + 4 to determine the
respect to y. Begin by sketching the cnrves and by solving y
limits of integration.
2—
phi
IA
U
::.-X
2
A
x
-
_.L
Limsc Tnfcfl
2
c:
3i-+
-
-
‘I—
1_.
1
=-I4
‘o1n
j-
/
Not 4
x
-
3 -4
jkI
3j
.jt
41
l
4
i-I
/L--_I’)
(4)
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2
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2
0
5. Simpson’s Rule and Velocity Curves
The following figure shoes velocity curves for two cars, A and B, that start side by side
and move along the same road. \Vhat does the area between the curves represent? Use
Simpson’s Rule to estimate it.
(A table of function values is provided to simplify the calculation)
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