Engineering Analysis

Engineering Analysis
1. Let ?? be the line (a counter clockwise circle centered at the origin with a
radius 1) defined in Figure 1. Its parametric representation is
???(??) = [
cos ??
sin ??
], 0 = ?? = 2??
??
Figure 1 Line C.
– Compute the line integral ? ??1???? + ??2???? ??
, where ??1 = 2?? and ??2 = 3??.
Hint: ? ??1???? + ??2???? ??
is the same as the line integral ? ??? · ?????
??
, where ??? =
[
??1
??2
]. Use sin2
?? =
1
2
(1 – cos 2??) and cos2
?? =
1
2
(1 + cos 2??).
– Re-compute this line integral using Green’s Theorem. Hint: rewrite the line
integral into double integral.
??
?? 1
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2. Let ??? = 0??? + ????? + 0???? and the surface ? be defined in Figure 2 as
?? + ?? + ?? = 1
?? = 0, ?? = 0, ?? = 0.
The parametric representation of the surface is
???(??, ??) = [
??
??
1 – ?? – ??
], 0 = ?? = ??, 0 = ?? = 1
Figure 2. Surface ? defined by ?? + ?? + ?? = 1, ?? = 0, ?? = 0, ?? = 0.
– Compute ???? =
?????(??,??)
????
×
?????(??,??)
???? and ???? =
????
|????|
.
– Compute the surface integral ? ? × ?? · ???? ????.
?
??
??
??
1
1
1
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– Using Stokes’ Theorem to re-compute the surface integral. Hint: rewrite the
surface integral into line integral. You need to find the lines (with
directions), develop theirs parametric form, integrate individually, then
compute the sum. Follow solutions to in-class practice on Oct. 21.
3. A typical application of triple integral is to compute the central of gravity. For
example, given a cylinder with its radius being 1 and height being 1, as shown in
Figure 3, and its density ??(??, ??, ??) = 1. The central of gravity is given by
(
? ????(??, ??, ??)???? ??
? ??(??, ??, ??)???? ??
,
? ????(??, ??, ??)???? ??
? ??(??, ??, ??)???? ??
,
? ????(??, ??, ??)???? ??
? ??(??, ??, ??)???? ??
),
where ?? is the cylinder. Compute ? ????(??,??,??)???? ??
? ??(??,??,??)???? ??
,
? ????(??,??,??)???? ??
? ??(??,??,??)???? ??
,
? ????(??,??,??)???? ??
? ??(??,??,??)???? ??
.
Figure 3 A cylinder with radius 1 and height 1.
??
1
1
??
??
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4. Let ??? = ????? + 0??? + 0????. Given a cylinder with its radius 1 and height 2, shown in
Figure 4.
Figure 4 A cylinder with radius 1 and height 2.
– Compute the triple integral ? ? · ??? ???? ??
, where ?? is the cylinder in Figure
4. Hint: compute the divergence of ?? and then compute triple integral.
– Rewrite the triple integral into surface integral using Divergence Theorem.
Hint: The surface includes the top, bottom, and side of the cylinder.
Top: ???(??, ??) = [
?? cos ??
?? sin ??
2
], 0 = ?? = 1, 0 = ?? = 2??,
|
?????(??, ??)
???? ×
?????(??, ??)
???? | = ??, ???? = [


1
]
2
1
??
??
??
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Bottom: ???(??, ??) = [
?? cos ??
?? sin ??

], 0 = ?? = 1, 0 = ?? = 2??,
|
?????(??, ??)
???? ×
?????(??, ??)
???? | = ??, ???? = [


-1
]
Side: ???(??, ??) = [
cos ??
sin ??
??
], 0 = ?? = 2, 0 = ?? = 2??,
|
?????(??, ??)
???? ×
?????(??, ??)
???? | = 1, ???? = [
cos ??
sin ??

]
Compute the surface integral individually (top, bottom, and side) and then
compute the sum. You may need to use cos2 ?? =
1
2
(1 + cos 2??).