volumes of solids of revolution · volumes of solids of revolution examples: find the volume of the...
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VolumesofSolidsofRevolution
Examples:FindthevolumeofthesolidobtainedbyrotatingtheregionRenclosed(bounded)bythegivencurvesaboutthegivenaxis.(a)(b)
€
y =1x
, y = 0, x =1, and x = 2 about the x - axis
€
y = 8 − x, y = 3, x = 2, and x = 5 about the y - axis
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Definite(Proper)Integrals
Assumptions:fiscontinuousonafiniteinterval[a,b].
€
f (x)dxa
b
∫
properintegral finiteregion
=realnumber
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ImproperIntegrals
Whyarethefollowingdefiniteintegrals“improper”?
€
1x 2dx
1
∞
∫
€
1xdx
0
4
∫
e−5x dx−∞
4
∫
1(x − 2)2
dx1
4
∫
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ImproperIntegralsTypeI:InfiniteLimitsofIntegration
Definition:Assumethatthedefiniteintegralexists(i.e.,isequaltoarealnumber)foreveryThenwedefinetheimproperintegraloff(x)onbyprovidedthatthelimitontherightsideexists. €
f (x)dxa
∞
∫ = limT→∞
f (x)dxa
T
∫⎛
⎝ ⎜
⎞
⎠ ⎟ €
T ≥ a.
€
f (x)dxa
T
∫
€
(a, ∞)
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ImproperIntegralsTypeI:InfiniteLimitsofIntegration
Illustration:
€
f (x)dxa
∞
∫ = limT→∞
f (x)dxa
T
∫⎛
⎝ ⎜
⎞
⎠ ⎟
properintegral
finiteregion
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ImproperIntegralsTypeI:InfiniteLimitsofIntegration
Examples:Evaluatethefollowingimproperintegrals.(a) (b)
€
1x 2dx
1
∞
∫
€
1xdx
1
∞
∫
![Page 7: Volumes of Solids of Revolution · Volumes of Solids of Revolution Examples: Find the volume of the solid obtained by rotating the region R enclosed (bounded) by the given curves](https://reader030.vdocuments.net/reader030/viewer/2022040215/5ecd1ad5afdb5c67276c99c6/html5/thumbnails/7.jpg)
ImproperIntegralsTypeI:InfiniteLimitsofIntegration
Whenthelimitexists,wesaythattheintegralconverges.
Whenthelimitdoesnotexist,wesaythattheintegraldiverges.
€
1x pdx
1
∞
∫Rule: isconvergentifanddivergentif
€
p >1
€
p ≤1
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IllustrationY
X�
Y
X�
€
1xdx
1
∞
∫
€
1x 2dx
1
∞
∫
infiniteareafinitearea
convergesdiverges
€
y =1x
€
y =1x 2
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ImproperIntegralsTypeII:InfiniteIntegrands
Definition:Assumethatf(x)iscontinuouson(a,b]butnotcontinuousatx=a.Thenwedefineprovidedthatthelimitontherightsideexists.
€
f (x)dxa
b
∫ = limT→ a +
f (x)dxT
b
∫
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ImproperIntegralsTypeII:InfiniteIntegrands
Illustration:
properintegral
finiteregion
f (x)dxa
b
∫ = limT→a+
f (x)dxT
b
∫⎛
⎝⎜
⎞
⎠⎟
y
x a b T
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ImproperIntegralsTypeII:InfiniteIntegrands
Examples:Evaluatethefollowingimproperintegrals.(a) (b)(c)
€
1x 2dx
0
10
∫
€
1x3 dx
0
2
∫
€
ln xxdx
0
1
∫
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ImproperIntegralsTypeII:InfiniteIntegrands
Whenthelimitexists,wesaythattheintegralconverges.
Whenthelimitdoesnotexist,wesaythattheintegraldiverges.
€
1x pdx
0
1
∫Rule: isconvergentifanddivergentif
€
0 < p <1
€
p ≥1
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Illustration
1x2dx
0
1
∫ 1x1/3
dx0
1
∫
infinitearea
finitearea
convergesdiverges
y = 1x2
y = 1x1/3
y
x 1 0
y
x 1