Definite (Proper) Integrals Assumptions : f is continuous on a finite interval [a,b]. f ( x ) dx a b ∫ proper integral finite region = real number
Definite(Proper)Integrals
Assumptions:fiscontinuousonafiniteinterval[a,b].
€
f (x)dxa
b
∫
properintegral finiteregion
=realnumber
ImproperIntegrals
Whyarethefollowingdefiniteintegrals“improper”?
€
1x 2dx
1
∞
∫
€
1xdx
0
4
∫
e−5x dx−∞
4
∫
1(x − 2)2
dx1
4
∫
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, ∞)
ImproperIntegralsTypeI:InfiniteLimitsofIntegration
Illustration:
€
f (x)dxa
∞
∫ = limT→∞
f (x)dxa
T
∫⎛
⎝ ⎜
⎞
⎠ ⎟
properintegral
finiteregion
ImproperIntegralsTypeI:InfiniteLimitsofIntegration
Examples:Evaluatethefollowingimproperintegrals.(a) (b)
€
1x 2dx
1
∞
∫
€
1xdx
1
∞
∫
ImproperIntegralsTypeI:InfiniteLimitsofIntegration
Whenthelimitexists,wesaythattheintegralconverges.
Whenthelimitdoesnotexist,wesaythattheintegraldiverges.
€
1x pdx
1
∞
∫Rule: isconvergentifanddivergentif
€
p >1
€
p ≤1
IllustrationY
X�
Y
X�
€
1xdx
1
∞
∫
€
1x 2dx
1
∞
∫
infiniteareafinitearea
convergesdiverges
€
y =1x
€
y =1x 2
Application
Example:p.584,#35.TheconcentrationofatoxininacellisincreasingatarateofstartingfromaconcentrationofIfthecellispoisonedwhentheconcentrationexceedscouldthiscellsurvive?€
50e−2t µmol /L /s,
€
10µmol /L.
€
30µmol /L,
ImproperIntegralsTypeII:InfiniteIntegrands
Definition:Assumethatf(x)iscontinuouson(a,b]butnotcontinuousatx=a.Thenwedefineprovidedthatthelimitontherightsideexists.
€
f (x)dxa
b
∫ = limT→ a +
f (x)dxT
b
∫
ImproperIntegralsTypeII:InfiniteIntegrands
Illustration:
properintegral
finiteregion
f (x)dxa
b
∫ = limT→a+
f (x)dxT
b
∫⎛
⎝⎜
⎞
⎠⎟
y
x a b T
ImproperIntegralsTypeII:InfiniteIntegrands
Examples:Evaluatethefollowingimproperintegrals.(a) (b)(c)
€
1x 2dx
0
10
∫
€
1x3 dx
0
2
∫
€
ln xxdx
0
1
∫
ImproperIntegralsTypeII:InfiniteIntegrands
Whenthelimitexists,wesaythattheintegralconverges.
Whenthelimitdoesnotexist,wesaythattheintegraldiverges.
€
1x pdx
0
1
∫Rule: isconvergentifanddivergentif
€
0 < p <1
€
p ≥1
Illustration
1x2dx
0
1
∫ 1x1/3
dx0
1
∫
infinitearea
finitearea
convergesdiverges
y = 1x2
y = 1x1/3
y
x 1 0
y
x 1