Indefinite Integrals and the Net Change Theorem · 05/09/2017  · Be careful to distinguish...

IndefiniteIntegrals:

∫ 𝑓(𝑥)𝑑𝑥'( canbeevaluatedas𝐹(𝑏) − 𝐹(𝑎).

Butwhatdoes∫ 𝑓(𝑥)𝑑𝑥�� mean?

A 𝑓(𝑥)𝑑𝑥

�

�

meansthatF(x),theanti-derivativeof𝑓(𝑥) hasaderivativeequalto𝑓(𝑥).Inotherwords,

A 𝑓(𝑥)𝑑𝑥

�

�

= 𝐹(𝑥)

means

𝐹L(𝑥) = 𝑓(𝑥)

Forexample,

A 𝑥N𝑑𝑥

�

�

=𝑥O

3⎯⎯⎯+ 𝐶

because𝑑𝑑𝑥⎯⎯⎯T𝑥O

3⎯⎯⎯+ 𝐶U = 𝑥N

Becarefultodistinguishdefiniteandindefiniteintegrals:

Thedefiniteintegral

A 𝑓(𝑥)𝑑𝑥

'

(isanumber;

thedefiniteintegral

A 𝑓(𝑥)𝑑𝑥

�

�isafunction.

Someindefiniteintegrals(anti-derivatives)andsomeproperties:

Examples:Findthegeneralintegrals:

A(10𝑥^ − 2𝑠𝑒𝑐N𝑥)𝑑𝑥

�

�

A𝑐𝑜𝑠𝜃𝑠𝑖𝑛N𝜃⎯⎯⎯⎯⎯ 𝑑𝜃

�

�(Hint:useatrigonometric identity)

A(𝑥O − 6𝑥)O

j

𝑑𝑥

TheNetChangeTheorem:Since

A 𝑓(𝑥)𝑑𝑥

�

�

= 𝐹(𝑥)

and

A 𝑓(𝑥)𝑑𝑥

'

(

= 𝐹(𝑏) − 𝐹(𝑎)

wecanwriteinstead,

𝐹′(𝑥) representsthe"slope"(thechangeinywithrespecttothechangein𝑝)ofourfunctionatapoint𝑥 = 𝑎,and𝐹(𝑏) − 𝐹(𝑎) isthenetchangeinthefunctionbetween𝑥 = 𝑎 and𝑥 = 𝑏.Netchangeisjustthetotalchangebetweenthetwovalues;if𝐹 isincreasing,thenthenetchangeisthesameastheaveragechange,butif𝐹 increasesandthendecreasesagainovertheintervalofinterest,thenetchangeistheaccumulatedpositiveandnegativechangeoverthatinterval.

Sincevelocity isthechangeinpositionwithrespecttothechangeintime,𝑣(𝑡) = 𝑠′(𝑡) (𝑠 representsposition).

Therefore,

A 𝑣(𝑡)𝑑𝑡

'

(

= 𝑉(𝑏) − 𝑉(𝑎)

or

A 𝑣(𝑡)𝑑𝑡

st

su

= 𝑉(𝑡N)− 𝑉(𝑡v)= 𝑠(𝑡N)− 𝑠(𝑡v)

Inotherwords,tofindthenetchangeinposition ofaparticlebetween𝑡vand𝑡N,evaluatetheintegralofitsvelocityfunctionoverthatinterval.

Iftherearetimesthattheposition oftheparticleismovinginanegativedirection, itschangeinpositionwillstillbeapositivenumericalvalue,soweintegratetheabsolutevalueofthevelocityfunction:

∫ |𝑣(𝑡)|𝑑𝑡stsu

= totaldistancetraveled.

Example:Aparticlemovesalongalinesothatitsvelocityattime𝑡is𝑣(𝑡) = 𝑡N − 𝑡 −6 (measuredinmeterspersecond).(a)Findthedisplacementoftheparticleduringthetimeperiod1 ≤ 𝑡 ≤ 4.(b)Findthedistancetraveledduringthistimeperiod.

Answer:(a)

A 𝑣(𝑡)𝑑𝑡

st

su

= 𝑠(𝑡N)− 𝑠(𝑡v)

becomes

A(𝑡N−𝑡− 6)𝑑𝑡

^

v

= 𝑠(1) − 𝑠(4) =𝑡O

3⎯⎯⎯−𝑡N

2⎯⎯⎯− 6𝑡{

v

^

= |4O

3⎯⎯⎯−4N

2⎯⎯⎯− 6(4)} − |

1O

3⎯⎯⎯−1N

2⎯⎯⎯− 6(1)} = −

92⎯⎯

Theparticlehasmovedtotheleft(negative)fromitsoriginalposition 4.5meters.

(b)Thetotaldistancetraveledisthesumofthedistancemovedtotherightandtotheleft.Notethatthepositionfunction𝑡N − 𝑡 − 6 = (t − 3)(t+ 2) ispositiveontheinterval[3,4] andnegativeontheinterval[1,3],sothedistancetraveledis

A |𝑣(𝑡)|𝑑𝑡

^

v

= A(𝑡N−𝑡 − 6)𝑑𝑡 +A(𝑡N−𝑡 − 6)𝑑𝑡

^

O

O

v

=𝑡O

3⎯⎯⎯−𝑡N

2⎯⎯⎯− 6𝑡{

v

O

+𝑡O

3⎯⎯⎯−𝑡N

2⎯⎯⎯− 6𝑡{

O

^

= �v�⎯⎯≈ 10.17m.

Homework:#1-11odd,15,17,21-43odd,51,59

IndefiniteIntegralsandtheNetChangeTheorem

IndefiniteIntegrals:

∫ 𝑓(𝑥)𝑑𝑥'( canbeevaluatedas𝐹(𝑏) − 𝐹(𝑎).

Butwhatdoes∫ 𝑓(𝑥)𝑑𝑥�� mean?

A 𝑓(𝑥)𝑑𝑥

�

�

meansthatF(x),theanti-derivativeof𝑓(𝑥) hasaderivativeequalto𝑓(𝑥).Inotherwords,

A 𝑓(𝑥)𝑑𝑥

�

�

= 𝐹(𝑥)

means

𝐹L(𝑥) = 𝑓(𝑥)

Forexample,

A 𝑥N𝑑𝑥

�

�

=𝑥O

3⎯⎯⎯+ 𝐶

because𝑑𝑑𝑥⎯⎯⎯T𝑥O

3⎯⎯⎯+ 𝐶U = 𝑥N

Becarefultodistinguishdefiniteandindefiniteintegrals:

Thedefiniteintegral

A 𝑓(𝑥)𝑑𝑥

'

(isanumber;

thedefiniteintegral

A 𝑓(𝑥)𝑑𝑥

�

�isafunction.

Someindefiniteintegrals(anti-derivatives)andsomeproperties:

Examples:Findthegeneralintegrals:

A(10𝑥^ − 2𝑠𝑒𝑐N𝑥)𝑑𝑥

�

�

A𝑐𝑜𝑠𝜃𝑠𝑖𝑛N𝜃⎯⎯⎯⎯⎯ 𝑑𝜃

�

�(Hint:useatrigonometric identity)

A(𝑥O − 6𝑥)O

j

𝑑𝑥

TheNetChangeTheorem:Since

A 𝑓(𝑥)𝑑𝑥

�

�

= 𝐹(𝑥)

and

A 𝑓(𝑥)𝑑𝑥

'

(

= 𝐹(𝑏) − 𝐹(𝑎)

wecanwriteinstead,

𝐹′(𝑥) representsthe"slope"(thechangeinywithrespecttothechangein𝑝)ofourfunctionatapoint𝑥 = 𝑎,and𝐹(𝑏) − 𝐹(𝑎) isthenetchangeinthefunctionbetween𝑥 = 𝑎 and𝑥 = 𝑏.Netchangeisjustthetotalchangebetweenthetwovalues;if𝐹 isincreasing,thenthenetchangeisthesameastheaveragechange,butif𝐹 increasesandthendecreasesagainovertheintervalofinterest,thenetchangeistheaccumulatedpositiveandnegativechangeoverthatinterval.

Sincevelocity isthechangeinpositionwithrespecttothechangeintime,𝑣(𝑡) = 𝑠′(𝑡) (𝑠 representsposition).

Therefore,

A 𝑣(𝑡)𝑑𝑡

'

(

= 𝑉(𝑏) − 𝑉(𝑎)

or

A 𝑣(𝑡)𝑑𝑡

st

su

= 𝑉(𝑡N)− 𝑉(𝑡v)= 𝑠(𝑡N)− 𝑠(𝑡v)

Inotherwords,tofindthenetchangeinposition ofaparticlebetween𝑡vand𝑡N,evaluatetheintegralofitsvelocityfunctionoverthatinterval.

Iftherearetimesthattheposition oftheparticleismovinginanegativedirection, itschangeinpositionwillstillbeapositivenumericalvalue,soweintegratetheabsolutevalueofthevelocityfunction:

∫ |𝑣(𝑡)|𝑑𝑡stsu

= totaldistancetraveled.

Example:Aparticlemovesalongalinesothatitsvelocityattime𝑡is𝑣(𝑡) = 𝑡N − 𝑡 −6 (measuredinmeterspersecond).(a)Findthedisplacementoftheparticleduringthetimeperiod1 ≤ 𝑡 ≤ 4.(b)Findthedistancetraveledduringthistimeperiod.

Answer:(a)

A 𝑣(𝑡)𝑑𝑡

st

su

= 𝑠(𝑡N)− 𝑠(𝑡v)

becomes

A(𝑡N−𝑡− 6)𝑑𝑡

^

v

= 𝑠(1) − 𝑠(4) =𝑡O

3⎯⎯⎯−𝑡N

2⎯⎯⎯− 6𝑡{

v

^

= |4O

3⎯⎯⎯−4N

2⎯⎯⎯− 6(4)} − |

1O

3⎯⎯⎯−1N

2⎯⎯⎯− 6(1)} = −

92⎯⎯

Theparticlehasmovedtotheleft(negative)fromitsoriginalposition 4.5meters.

(b)Thetotaldistancetraveledisthesumofthedistancemovedtotherightandtotheleft.Notethatthepositionfunction𝑡N − 𝑡 − 6 = (t − 3)(t+ 2) ispositiveontheinterval[3,4] andnegativeontheinterval[1,3],sothedistancetraveledis

A |𝑣(𝑡)|𝑑𝑡

^

v

= A(𝑡N−𝑡 − 6)𝑑𝑡 +A(𝑡N−𝑡 − 6)𝑑𝑡

^

O

O

v

=𝑡O

3⎯⎯⎯−𝑡N

2⎯⎯⎯− 6𝑡{

v

O

+𝑡O

3⎯⎯⎯−𝑡N

2⎯⎯⎯− 6𝑡{

O

^

= �v�⎯⎯≈ 10.17m.

Homework:#1-11odd,15,17,21-43odd,51,59

IndefiniteIntegralsandtheNetChangeTheorem

IndefiniteIntegrals:

∫ 𝑓(𝑥)𝑑𝑥'( canbeevaluatedas𝐹(𝑏) − 𝐹(𝑎).

Butwhatdoes∫ 𝑓(𝑥)𝑑𝑥�� mean?

A 𝑓(𝑥)𝑑𝑥

�

�

meansthatF(x),theanti-derivativeof𝑓(𝑥) hasaderivativeequalto𝑓(𝑥).Inotherwords,

A 𝑓(𝑥)𝑑𝑥

�

�

= 𝐹(𝑥)

means

𝐹L(𝑥) = 𝑓(𝑥)

Forexample,

A 𝑥N𝑑𝑥

�

�

=𝑥O

3⎯⎯⎯+ 𝐶

because𝑑𝑑𝑥⎯⎯⎯T𝑥O

3⎯⎯⎯+ 𝐶U = 𝑥N

Becarefultodistinguishdefiniteandindefiniteintegrals:

Thedefiniteintegral

A 𝑓(𝑥)𝑑𝑥

'

(isanumber;

thedefiniteintegral

A 𝑓(𝑥)𝑑𝑥

�

�isafunction.

Someindefiniteintegrals(anti-derivatives)andsomeproperties:

Examples:Findthegeneralintegrals:

A(10𝑥^ − 2𝑠𝑒𝑐N𝑥)𝑑𝑥

�

�

A𝑐𝑜𝑠𝜃𝑠𝑖𝑛N𝜃⎯⎯⎯⎯⎯ 𝑑𝜃

�

�(Hint:useatrigonometric identity)

A(𝑥O − 6𝑥)O

j

𝑑𝑥

TheNetChangeTheorem:Since

A 𝑓(𝑥)𝑑𝑥

�

�

= 𝐹(𝑥)

and

A 𝑓(𝑥)𝑑𝑥

'

(

= 𝐹(𝑏) − 𝐹(𝑎)

wecanwriteinstead,

𝐹′(𝑥) representsthe"slope"(thechangeinywithrespecttothechangein𝑝)ofourfunctionatapoint𝑥 = 𝑎,and𝐹(𝑏) − 𝐹(𝑎) isthenetchangeinthefunctionbetween𝑥 = 𝑎 and𝑥 = 𝑏.Netchangeisjustthetotalchangebetweenthetwovalues;if𝐹 isincreasing,thenthenetchangeisthesameastheaveragechange,butif𝐹 increasesandthendecreasesagainovertheintervalofinterest,thenetchangeistheaccumulatedpositiveandnegativechangeoverthatinterval.

Sincevelocity isthechangeinpositionwithrespecttothechangeintime,𝑣(𝑡) = 𝑠′(𝑡) (𝑠 representsposition).

Therefore,

A 𝑣(𝑡)𝑑𝑡

'

(

= 𝑉(𝑏) − 𝑉(𝑎)

or

A 𝑣(𝑡)𝑑𝑡

st

su

= 𝑉(𝑡N)− 𝑉(𝑡v)= 𝑠(𝑡N)− 𝑠(𝑡v)

Inotherwords,tofindthenetchangeinposition ofaparticlebetween𝑡vand𝑡N,evaluatetheintegralofitsvelocityfunctionoverthatinterval.

Iftherearetimesthattheposition oftheparticleismovinginanegativedirection, itschangeinpositionwillstillbeapositivenumericalvalue,soweintegratetheabsolutevalueofthevelocityfunction:

∫ |𝑣(𝑡)|𝑑𝑡stsu

= totaldistancetraveled.

Example:Aparticlemovesalongalinesothatitsvelocityattime𝑡is𝑣(𝑡) = 𝑡N − 𝑡 −6 (measuredinmeterspersecond).(a)Findthedisplacementoftheparticleduringthetimeperiod1 ≤ 𝑡 ≤ 4.(b)Findthedistancetraveledduringthistimeperiod.

Answer:(a)

A 𝑣(𝑡)𝑑𝑡

st

su

= 𝑠(𝑡N)− 𝑠(𝑡v)

becomes

A(𝑡N−𝑡− 6)𝑑𝑡

^

v

= 𝑠(1) − 𝑠(4) =𝑡O

3⎯⎯⎯−𝑡N

2⎯⎯⎯− 6𝑡{

v

^

= |4O

3⎯⎯⎯−4N

2⎯⎯⎯− 6(4)} − |

1O

3⎯⎯⎯−1N

2⎯⎯⎯− 6(1)} = −

92⎯⎯

Theparticlehasmovedtotheleft(negative)fromitsoriginalposition 4.5meters.

(b)Thetotaldistancetraveledisthesumofthedistancemovedtotherightandtotheleft.Notethatthepositionfunction𝑡N − 𝑡 − 6 = (t − 3)(t+ 2) ispositiveontheinterval[3,4] andnegativeontheinterval[1,3],sothedistancetraveledis

A |𝑣(𝑡)|𝑑𝑡

^

v

= A(𝑡N−𝑡 − 6)𝑑𝑡 +A(𝑡N−𝑡 − 6)𝑑𝑡

^

O

O

v

=𝑡O

3⎯⎯⎯−𝑡N

2⎯⎯⎯− 6𝑡{

v

O

+𝑡O

3⎯⎯⎯−𝑡N

2⎯⎯⎯− 6𝑡{

O

^

= �v�⎯⎯≈ 10.17m.

Homework:#1-11odd,15,17,21-43odd,51,59

IndefiniteIntegralsandtheNetChangeTheorem

IndefiniteIntegrals:

∫ 𝑓(𝑥)𝑑𝑥'( canbeevaluatedas𝐹(𝑏) − 𝐹(𝑎).

Butwhatdoes∫ 𝑓(𝑥)𝑑𝑥�� mean?

A 𝑓(𝑥)𝑑𝑥

�

�

meansthatF(x),theanti-derivativeof𝑓(𝑥) hasaderivativeequalto𝑓(𝑥).Inotherwords,

A 𝑓(𝑥)𝑑𝑥

�

�

= 𝐹(𝑥)

means

𝐹L(𝑥) = 𝑓(𝑥)

Forexample,

A 𝑥N𝑑𝑥

�

�

=𝑥O

3⎯⎯⎯+ 𝐶

because𝑑𝑑𝑥⎯⎯⎯T𝑥O

3⎯⎯⎯+ 𝐶U = 𝑥N

Becarefultodistinguishdefiniteandindefiniteintegrals:

Thedefiniteintegral

A 𝑓(𝑥)𝑑𝑥

'

(isanumber;

thedefiniteintegral

A 𝑓(𝑥)𝑑𝑥

�

�isafunction.

Someindefiniteintegrals(anti-derivatives)andsomeproperties:

Examples:Findthegeneralintegrals:

A(10𝑥^ − 2𝑠𝑒𝑐N𝑥)𝑑𝑥

�

�

A𝑐𝑜𝑠𝜃𝑠𝑖𝑛N𝜃⎯⎯⎯⎯⎯ 𝑑𝜃

�

�(Hint:useatrigonometric identity)

A(𝑥O − 6𝑥)O

j

𝑑𝑥

TheNetChangeTheorem:Since

A 𝑓(𝑥)𝑑𝑥

�

�

= 𝐹(𝑥)

and

A 𝑓(𝑥)𝑑𝑥

'

(

= 𝐹(𝑏) − 𝐹(𝑎)

wecanwriteinstead,

𝐹′(𝑥) representsthe"slope"(thechangeinywithrespecttothechangein𝑝)ofourfunctionatapoint𝑥 = 𝑎,and𝐹(𝑏) − 𝐹(𝑎) isthenetchangeinthefunctionbetween𝑥 = 𝑎 and𝑥 = 𝑏.Netchangeisjustthetotalchangebetweenthetwovalues;if𝐹 isincreasing,thenthenetchangeisthesameastheaveragechange,butif𝐹 increasesandthendecreasesagainovertheintervalofinterest,thenetchangeistheaccumulatedpositiveandnegativechangeoverthatinterval.

Sincevelocity isthechangeinpositionwithrespecttothechangeintime,𝑣(𝑡) = 𝑠′(𝑡) (𝑠 representsposition).

Therefore,

A 𝑣(𝑡)𝑑𝑡

'

(

= 𝑉(𝑏) − 𝑉(𝑎)

or

A 𝑣(𝑡)𝑑𝑡

st

su

= 𝑉(𝑡N)− 𝑉(𝑡v)= 𝑠(𝑡N)− 𝑠(𝑡v)

Inotherwords,tofindthenetchangeinposition ofaparticlebetween𝑡vand𝑡N,evaluatetheintegralofitsvelocityfunctionoverthatinterval.

Iftherearetimesthattheposition oftheparticleismovinginanegativedirection, itschangeinpositionwillstillbeapositivenumericalvalue,soweintegratetheabsolutevalueofthevelocityfunction:

∫ |𝑣(𝑡)|𝑑𝑡stsu

= totaldistancetraveled.

Example:Aparticlemovesalongalinesothatitsvelocityattime𝑡is𝑣(𝑡) = 𝑡N − 𝑡 −6 (measuredinmeterspersecond).(a)Findthedisplacementoftheparticleduringthetimeperiod1 ≤ 𝑡 ≤ 4.(b)Findthedistancetraveledduringthistimeperiod.

Answer:(a)

A 𝑣(𝑡)𝑑𝑡

st

su

= 𝑠(𝑡N)− 𝑠(𝑡v)

becomes

A(𝑡N−𝑡− 6)𝑑𝑡

^

v

= 𝑠(1) − 𝑠(4) =𝑡O

3⎯⎯⎯−𝑡N

2⎯⎯⎯− 6𝑡{

v

^

= |4O

3⎯⎯⎯−4N

2⎯⎯⎯− 6(4)} − |

1O

3⎯⎯⎯−1N

2⎯⎯⎯− 6(1)} = −

92⎯⎯

Theparticlehasmovedtotheleft(negative)fromitsoriginalposition 4.5meters.

(b)Thetotaldistancetraveledisthesumofthedistancemovedtotherightandtotheleft.Notethatthepositionfunction𝑡N − 𝑡 − 6 = (t − 3)(t+ 2) ispositiveontheinterval[3,4] andnegativeontheinterval[1,3],sothedistancetraveledis

A |𝑣(𝑡)|𝑑𝑡

^

v

= A(𝑡N−𝑡 − 6)𝑑𝑡 +A(𝑡N−𝑡 − 6)𝑑𝑡

^

O

O

v

=𝑡O

3⎯⎯⎯−𝑡N

2⎯⎯⎯− 6𝑡{

v

O

+𝑡O

3⎯⎯⎯−𝑡N

2⎯⎯⎯− 6𝑡{

O

^

= �v�⎯⎯≈ 10.17m.

Homework:#1-11odd,15,17,21-43odd,51,59

IndefiniteIntegralsandtheNetChangeTheorem

IndefiniteIntegrals:

∫ 𝑓(𝑥)𝑑𝑥'( canbeevaluatedas𝐹(𝑏) − 𝐹(𝑎).

Butwhatdoes∫ 𝑓(𝑥)𝑑𝑥�� mean?

A 𝑓(𝑥)𝑑𝑥

�

�

meansthatF(x),theanti-derivativeof𝑓(𝑥) hasaderivativeequalto𝑓(𝑥).Inotherwords,

A 𝑓(𝑥)𝑑𝑥

�

�

= 𝐹(𝑥)

means

𝐹L(𝑥) = 𝑓(𝑥)

Forexample,

A 𝑥N𝑑𝑥

�

�

=𝑥O

3⎯⎯⎯+ 𝐶

because𝑑𝑑𝑥⎯⎯⎯T𝑥O

3⎯⎯⎯+ 𝐶U = 𝑥N

Becarefultodistinguishdefiniteandindefiniteintegrals:

Thedefiniteintegral

A 𝑓(𝑥)𝑑𝑥

'

(isanumber;

thedefiniteintegral

A 𝑓(𝑥)𝑑𝑥

�

�isafunction.

Someindefiniteintegrals(anti-derivatives)andsomeproperties:

Examples:Findthegeneralintegrals:

A(10𝑥^ − 2𝑠𝑒𝑐N𝑥)𝑑𝑥

�

�

A𝑐𝑜𝑠𝜃𝑠𝑖𝑛N𝜃⎯⎯⎯⎯⎯ 𝑑𝜃

�

�(Hint:useatrigonometric identity)

A(𝑥O − 6𝑥)O

j

𝑑𝑥

TheNetChangeTheorem:Since

A 𝑓(𝑥)𝑑𝑥

�

�

= 𝐹(𝑥)

and

A 𝑓(𝑥)𝑑𝑥

'

(

= 𝐹(𝑏) − 𝐹(𝑎)

wecanwriteinstead,

𝐹′(𝑥) representsthe"slope"(thechangeinywithrespecttothechangein𝑝)ofourfunctionatapoint𝑥 = 𝑎,and𝐹(𝑏) − 𝐹(𝑎) isthenetchangeinthefunctionbetween𝑥 = 𝑎 and𝑥 = 𝑏.Netchangeisjustthetotalchangebetweenthetwovalues;if𝐹 isincreasing,thenthenetchangeisthesameastheaveragechange,butif𝐹 increasesandthendecreasesagainovertheintervalofinterest,thenetchangeistheaccumulatedpositiveandnegativechangeoverthatinterval.

Sincevelocity isthechangeinpositionwithrespecttothechangeintime,𝑣(𝑡) = 𝑠′(𝑡) (𝑠 representsposition).

Therefore,

A 𝑣(𝑡)𝑑𝑡

'

(

= 𝑉(𝑏) − 𝑉(𝑎)

or

A 𝑣(𝑡)𝑑𝑡

st

su

= 𝑉(𝑡N)− 𝑉(𝑡v)= 𝑠(𝑡N)− 𝑠(𝑡v)

Inotherwords,tofindthenetchangeinposition ofaparticlebetween𝑡vand𝑡N,evaluatetheintegralofitsvelocityfunctionoverthatinterval.

Iftherearetimesthattheposition oftheparticleismovinginanegativedirection, itschangeinpositionwillstillbeapositivenumericalvalue,soweintegratetheabsolutevalueofthevelocityfunction:

∫ |𝑣(𝑡)|𝑑𝑡stsu

= totaldistancetraveled.

Example:Aparticlemovesalongalinesothatitsvelocityattime𝑡is𝑣(𝑡) = 𝑡N − 𝑡 −6 (measuredinmeterspersecond).(a)Findthedisplacementoftheparticleduringthetimeperiod1 ≤ 𝑡 ≤ 4.(b)Findthedistancetraveledduringthistimeperiod.

Answer:(a)

A 𝑣(𝑡)𝑑𝑡

st

su

= 𝑠(𝑡N)− 𝑠(𝑡v)

becomes

A(𝑡N−𝑡− 6)𝑑𝑡

^

v

= 𝑠(1) − 𝑠(4) =𝑡O

3⎯⎯⎯−𝑡N

2⎯⎯⎯− 6𝑡{

v

^

= |4O

3⎯⎯⎯−4N

2⎯⎯⎯− 6(4)} − |

1O

3⎯⎯⎯−1N

2⎯⎯⎯− 6(1)} = −

92⎯⎯

Theparticlehasmovedtotheleft(negative)fromitsoriginalposition 4.5meters.

(b)Thetotaldistancetraveledisthesumofthedistancemovedtotherightandtotheleft.Notethatthepositionfunction𝑡N − 𝑡 − 6 = (t − 3)(t+ 2) ispositiveontheinterval[3,4] andnegativeontheinterval[1,3],sothedistancetraveledis

A |𝑣(𝑡)|𝑑𝑡

^

v

= A(𝑡N−𝑡 − 6)𝑑𝑡 +A(𝑡N−𝑡 − 6)𝑑𝑡

^

O

O

v

=𝑡O

3⎯⎯⎯−𝑡N

2⎯⎯⎯− 6𝑡{

v

O

+𝑡O

3⎯⎯⎯−𝑡N

2⎯⎯⎯− 6𝑡{

O

^

= �v�⎯⎯≈ 10.17m.

Homework:#1-11odd,15,17,21-43odd,51,59

IndefiniteIntegralsandtheNetChangeTheorem

IndefiniteIntegrals:

∫ 𝑓(𝑥)𝑑𝑥'( canbeevaluatedas𝐹(𝑏) − 𝐹(𝑎).

Butwhatdoes∫ 𝑓(𝑥)𝑑𝑥�� mean?

A 𝑓(𝑥)𝑑𝑥

�

�

meansthatF(x),theanti-derivativeof𝑓(𝑥) hasaderivativeequalto𝑓(𝑥).Inotherwords,

A 𝑓(𝑥)𝑑𝑥

�

�

= 𝐹(𝑥)

means

𝐹L(𝑥) = 𝑓(𝑥)

Forexample,

A 𝑥N𝑑𝑥

�

�

=𝑥O

3⎯⎯⎯+ 𝐶

because𝑑𝑑𝑥⎯⎯⎯T𝑥O

3⎯⎯⎯+ 𝐶U = 𝑥N

Becarefultodistinguishdefiniteandindefiniteintegrals:

Thedefiniteintegral

A 𝑓(𝑥)𝑑𝑥

'

(isanumber;

thedefiniteintegral

A 𝑓(𝑥)𝑑𝑥

�

�isafunction.

Someindefiniteintegrals(anti-derivatives)andsomeproperties:

Examples:Findthegeneralintegrals:

A(10𝑥^ − 2𝑠𝑒𝑐N𝑥)𝑑𝑥

�

�

A𝑐𝑜𝑠𝜃𝑠𝑖𝑛N𝜃⎯⎯⎯⎯⎯ 𝑑𝜃

�

�(Hint:useatrigonometric identity)

A(𝑥O − 6𝑥)O

j

𝑑𝑥

TheNetChangeTheorem:Since

A 𝑓(𝑥)𝑑𝑥

�

�

= 𝐹(𝑥)

and

A 𝑓(𝑥)𝑑𝑥

'

(

= 𝐹(𝑏) − 𝐹(𝑎)

wecanwriteinstead,

𝐹′(𝑥) representsthe"slope"(thechangeinywithrespecttothechangein𝑝)ofourfunctionatapoint𝑥 = 𝑎,and𝐹(𝑏) − 𝐹(𝑎) isthenetchangeinthefunctionbetween𝑥 = 𝑎 and𝑥 = 𝑏.Netchangeisjustthetotalchangebetweenthetwovalues;if𝐹 isincreasing,thenthenetchangeisthesameastheaveragechange,butif𝐹 increasesandthendecreasesagainovertheintervalofinterest,thenetchangeistheaccumulatedpositiveandnegativechangeoverthatinterval.

Sincevelocity isthechangeinpositionwithrespecttothechangeintime,𝑣(𝑡) = 𝑠′(𝑡) (𝑠 representsposition).

Therefore,

A 𝑣(𝑡)𝑑𝑡

'

(

= 𝑉(𝑏) − 𝑉(𝑎)

or

A 𝑣(𝑡)𝑑𝑡

st

su

= 𝑉(𝑡N)− 𝑉(𝑡v)= 𝑠(𝑡N)− 𝑠(𝑡v)

Inotherwords,tofindthenetchangeinposition ofaparticlebetween𝑡vand𝑡N,evaluatetheintegralofitsvelocityfunctionoverthatinterval.

Iftherearetimesthattheposition oftheparticleismovinginanegativedirection, itschangeinpositionwillstillbeapositivenumericalvalue,soweintegratetheabsolutevalueofthevelocityfunction:

∫ |𝑣(𝑡)|𝑑𝑡stsu

= totaldistancetraveled.

Example:Aparticlemovesalongalinesothatitsvelocityattime𝑡is𝑣(𝑡) = 𝑡N − 𝑡 −6 (measuredinmeterspersecond).(a)Findthedisplacementoftheparticleduringthetimeperiod1 ≤ 𝑡 ≤ 4.(b)Findthedistancetraveledduringthistimeperiod.

Answer:(a)

A 𝑣(𝑡)𝑑𝑡

st

su

= 𝑠(𝑡N)− 𝑠(𝑡v)

becomes

A(𝑡N−𝑡− 6)𝑑𝑡

^

v

= 𝑠(1) − 𝑠(4) =𝑡O

3⎯⎯⎯−𝑡N

2⎯⎯⎯− 6𝑡{

v

^

= |4O

3⎯⎯⎯−4N

2⎯⎯⎯− 6(4)} − |

1O

3⎯⎯⎯−1N

2⎯⎯⎯− 6(1)} = −

92⎯⎯

Theparticlehasmovedtotheleft(negative)fromitsoriginalposition 4.5meters.

(b)Thetotaldistancetraveledisthesumofthedistancemovedtotherightandtotheleft.Notethatthepositionfunction𝑡N − 𝑡 − 6 = (t − 3)(t+ 2) ispositiveontheinterval[3,4] andnegativeontheinterval[1,3],sothedistancetraveledis

A |𝑣(𝑡)|𝑑𝑡

^

v

= A(𝑡N−𝑡 − 6)𝑑𝑡 +A(𝑡N−𝑡 − 6)𝑑𝑡

^

O

O

v

=𝑡O

3⎯⎯⎯−𝑡N

2⎯⎯⎯− 6𝑡{

v

O

+𝑡O

3⎯⎯⎯−𝑡N

2⎯⎯⎯− 6𝑡{

O

^

= �v�⎯⎯≈ 10.17m.

Homework:#1-11odd,15,17,21-43odd,51,59

IndefiniteIntegralsandtheNetChangeTheorem

IndefiniteIntegrals:

∫ 𝑓(𝑥)𝑑𝑥'( canbeevaluatedas𝐹(𝑏) − 𝐹(𝑎).

Butwhatdoes∫ 𝑓(𝑥)𝑑𝑥�� mean?

A 𝑓(𝑥)𝑑𝑥

�

�

meansthatF(x),theanti-derivativeof𝑓(𝑥) hasaderivativeequalto𝑓(𝑥).Inotherwords,

A 𝑓(𝑥)𝑑𝑥

�

�

= 𝐹(𝑥)

means

𝐹L(𝑥) = 𝑓(𝑥)

Forexample,

A 𝑥N𝑑𝑥

�

�

=𝑥O

3⎯⎯⎯+ 𝐶

because𝑑𝑑𝑥⎯⎯⎯T𝑥O

3⎯⎯⎯+ 𝐶U = 𝑥N

Becarefultodistinguishdefiniteandindefiniteintegrals:

Thedefiniteintegral

A 𝑓(𝑥)𝑑𝑥

'

(isanumber;

thedefiniteintegral

A 𝑓(𝑥)𝑑𝑥

�

�isafunction.

Someindefiniteintegrals(anti-derivatives)andsomeproperties:

Examples:Findthegeneralintegrals:

A(10𝑥^ − 2𝑠𝑒𝑐N𝑥)𝑑𝑥

�

�

A𝑐𝑜𝑠𝜃𝑠𝑖𝑛N𝜃⎯⎯⎯⎯⎯ 𝑑𝜃

�

�(Hint:useatrigonometric identity)

A(𝑥O − 6𝑥)O

j

𝑑𝑥

TheNetChangeTheorem:Since

A 𝑓(𝑥)𝑑𝑥

�

�

= 𝐹(𝑥)

and

A 𝑓(𝑥)𝑑𝑥

'

(

= 𝐹(𝑏) − 𝐹(𝑎)

wecanwriteinstead,

𝐹′(𝑥) representsthe"slope"(thechangeinywithrespecttothechangein𝑝)ofourfunctionatapoint𝑥 = 𝑎,and𝐹(𝑏) − 𝐹(𝑎) isthenetchangeinthefunctionbetween𝑥 = 𝑎 and𝑥 = 𝑏.Netchangeisjustthetotalchangebetweenthetwovalues;if𝐹 isincreasing,thenthenetchangeisthesameastheaveragechange,butif𝐹 increasesandthendecreasesagainovertheintervalofinterest,thenetchangeistheaccumulatedpositiveandnegativechangeoverthatinterval.

Sincevelocity isthechangeinpositionwithrespecttothechangeintime,𝑣(𝑡) = 𝑠′(𝑡) (𝑠 representsposition).

Therefore,

A 𝑣(𝑡)𝑑𝑡

'

(

= 𝑉(𝑏) − 𝑉(𝑎)

or

A 𝑣(𝑡)𝑑𝑡

st

su

= 𝑉(𝑡N)− 𝑉(𝑡v)= 𝑠(𝑡N)− 𝑠(𝑡v)

Inotherwords,tofindthenetchangeinposition ofaparticlebetween𝑡vand𝑡N,evaluatetheintegralofitsvelocityfunctionoverthatinterval.

Iftherearetimesthattheposition oftheparticleismovinginanegativedirection, itschangeinpositionwillstillbeapositivenumericalvalue,soweintegratetheabsolutevalueofthevelocityfunction:

∫ |𝑣(𝑡)|𝑑𝑡stsu

= totaldistancetraveled.

Example:Aparticlemovesalongalinesothatitsvelocityattime𝑡is𝑣(𝑡) = 𝑡N − 𝑡 −6 (measuredinmeterspersecond).(a)Findthedisplacementoftheparticleduringthetimeperiod1 ≤ 𝑡 ≤ 4.(b)Findthedistancetraveledduringthistimeperiod.

Answer:(a)

A 𝑣(𝑡)𝑑𝑡

st

su

= 𝑠(𝑡N)− 𝑠(𝑡v)

becomes

A(𝑡N−𝑡− 6)𝑑𝑡

^

v

= 𝑠(1) − 𝑠(4) =𝑡O

3⎯⎯⎯−𝑡N

2⎯⎯⎯− 6𝑡{

v

^

= |4O

3⎯⎯⎯−4N

2⎯⎯⎯− 6(4)} − |

1O

3⎯⎯⎯−1N

2⎯⎯⎯− 6(1)} = −

92⎯⎯

Theparticlehasmovedtotheleft(negative)fromitsoriginalposition 4.5meters.

(b)Thetotaldistancetraveledisthesumofthedistancemovedtotherightandtotheleft.Notethatthepositionfunction𝑡N − 𝑡 − 6 = (t − 3)(t+ 2) ispositiveontheinterval[3,4] andnegativeontheinterval[1,3],sothedistancetraveledis

A |𝑣(𝑡)|𝑑𝑡

^

v

= A(𝑡N−𝑡 − 6)𝑑𝑡 +A(𝑡N−𝑡 − 6)𝑑𝑡

^

O

O

v

=𝑡O

3⎯⎯⎯−𝑡N

2⎯⎯⎯− 6𝑡{

v

O

+𝑡O

3⎯⎯⎯−𝑡N

2⎯⎯⎯− 6𝑡{

O

^

= �v�⎯⎯≈ 10.17m.

Homework:#1-11odd,15,17,21-43odd,51,59

IndefiniteIntegralsandtheNetChangeTheorem