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

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

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