warm-up: 1)if a particle has a velocity function defined by, find its acceleration function. 2)if a...
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Warm-up:1) If a particle has a velocity function
defined by , find its acceleration function.
2) If a particle has an acceleration function defined by , what is its velocity function? Is there more
than one possibility?
2563)( 34 ttttv
54)( 3 xta
IntegrationSection 6.1 & 6.2
The Area Under a Curve / Indefinite Integrals
The Rectangle Method for Finding Areas
• Read over Example on pg. 351 and draw a conclusion about the Area under of curve as it relates to the number of rectangular sub-intervals.
The Rectangle Method for Finding Areas
• Read over Example on pg. 351 and draw a conclusion about the Area under of curve as it relates to the number of rectangular sub-intervals.
• When we become more comfortable with integration we will use integrals to more accurately find the area under a curve.
Anti-differentiation (Integration)
• The opposite of derivatives (anti-derivatives)
• Ex: You are given a velocity function and you want to find out what the position function is for the particle. How would you determine s(t)?
Anti-differentiation (Integration)
• The opposite of derivatives (anti-derivatives)
• Ex: You are given a velocity function and you want to find out what the position function is for the particle. How would you determine s(t)?
• Let’s assume ttv 2
Anti-differentiation (Integration)
• The opposite of derivatives (anti-derivatives)
• Ex: You are given a velocity function and you want to find out what the position function is for the particle. How would you determine s(t)?
• Let’s assume
ttv 2
2tts
Anti-differentiation (Integration)
• The opposite of derivatives (anti-derivatives)
• Ex: You are given a velocity function and you want to find out what the position function is for the particle. How would you determine s(t)?
• Let’s assume
• Could work?
ttv 2
2tts
32 tts
Anti-differentiation (Integration)
• The opposite of derivatives (anti-derivatives)
• Ex: You are given a velocity function and you want to find out what the position function is for the particle. How would you determine s(t)?
• Let’s assume
• Could work? How about ?
ttv 2
2tts
32 tts 52 tts
Indefinite Integrals• The process of finding anti-derivatives is
called Anti-Differentiation or Integration.
•
Indefinite Integrals• The process of finding anti-derivatives is
called Anti-Differentiation or Integration.
• )()]([ xfxFdx
d
Indefinite Integrals• The process of finding anti-derivatives is
called Anti-Differentiation or Integration.
• can be written as using Integral Notation,
)()]([ xfxFdx
d CxFdxxf )()(
Indefinite Integrals• The process of finding anti-derivatives is
called Anti-Differentiation or Integration.
• can be written as using Integral Notation, where the expression is called an Indefinite Integral,
)()]([ xfxFdx
d CxFdxxf )()(
dxxf )(
Indefinite Integrals• The process of finding anti-derivatives is
called Anti-Differentiation or Integration.
• can be written as using Integral Notation, where the expression is called an Indefinite Integral, the function f(x) is called the Integrand,
)()]([ xfxFdx
d CxFdxxf )()(
dxxf )(
Indefinite Integrals• The process of finding anti-derivatives is
called Anti-Differentiation or Integration.
• can be written as using Integral Notation, where the expression is called an Indefinite Integral, the function f(x) is called the Integrand, and the constant C is called the Constant of Integration.
)()]([ xfxFdx
d CxFdxxf )()(
dxxf )(
Properties of Integrals:
Properties of Integrals:• A constant Factor can be moved through
an Integral sign:
Properties of Integrals:• A constant Factor can be moved through
an Integral sign:dxxfcdxxcf )()(
Properties of Integrals:• A constant Factor can be moved through
an Integral sign:
• An anti-derivative of a sum is the sum of the anti-derivatives:
dxxfcdxxcf )()(
Properties of Integrals:• A constant Factor can be moved through
an Integral sign:
• An anti-derivative of a sum is the sum of the anti-derivatives:
dxxfcdxxcf )()(
dxxgdxxfdxxgxf )()()]()([
Properties of Integrals:• A constant Factor can be moved through
an Integral sign:
• An anti-derivative of a sum is the sum of the anti-derivatives:
• An anti-derivative of a difference is the difference of the anti-derivatives:
dxxfcdxxcf )()(
dxxgdxxfdxxgxf )()()]()([
Properties of Integrals:• A constant Factor can be moved through
an Integral sign:
• An anti-derivative of a sum is the sum of the anti-derivatives:
• An anti-derivative of a difference is the difference of the anti-derivatives:
dxxfcdxxcf )()(
dxxgdxxfdxxgxf )()()]()([
dxxgdxxfdxxgxf )()()]()([
Integral Power Rule
• To integrate a power function (other than -1), add 1 to the exponent and divide by the new exponent.
Cr
xdxx
rr
1
1
Integral Power Rule
• To integrate a power function (other than -1), add 1 to the exponent and divide by the new exponent.
• Find
Cr
xdxx
rr
1
1
dxx23
Integral Power Rule
• To integrate a power function (other than -1), add 1 to the exponent and divide by the new exponent.
• Find
Cr
xdxx
rr
1
1
dxxdxx 22 33
Integral Power Rule
• To integrate a power function (other than -1), add 1 to the exponent and divide by the new exponent.
• Find
Cr
xdxx
rr
1
1
dxxdxx 22 33
Cx
)3(3
3
Integral Power Rule
• To integrate a power function (other than -1), add 1 to the exponent and divide by the new exponent.
• Find
Cr
xdxx
rr
1
1
dxxdxx 22 33
Cx
)3(3
3
Cx 3
Examples (S)1) Find
2) Find
3) Find
dxx2
dxx41
dxx
Examples (S)1) Find
2) Find
3) Find
Cx
dxx 3
32
dxx41
dxx
Examples1) Find
2) Find
3) Find
Cx
dxx 3
32
dxxdxx 44
1
dxx
Examples1) Find
2) Find
3) Find
Cx
dxx 3
32
Cx
dxxdxx
3
1 34
4
dxx
Examples1) Find
2) Find
3) Find
Cx
dxx 3
32
Cx
Cx
dxxdxx
3
34
4 3
1
3
1
dxx
Examples1) Find
2) Find
3) Find
Cx
dxx 3
32
Cx
Cx
dxxdxx
3
34
4 3
1
3
1
dxxdxx 2
1
Examples1) Find
2) Find
3) Find
Cx
dxx 3
32
Cx
Cx
dxxdxx
3
34
4 3
1
3
1
Cx
dxxdxx 23
2
3
2
1
Examples1) Find
2) Find
3) Find
Cx
dxx 3
32
Cx
Cx
dxxdxx
3
34
4 3
1
3
1
Cx
Cx
dxxdxx 3
2
23
2
3
2
3
2
1
Examples of Common Integrals
1) Find
2) Find
dxxcos
dx
x21
1
Examples of Common Integrals
1) Find
2) Find
dxxcos
dx
x21
1
Cx sin
Examples of Common Integrals
1) Find
2) Find
dxxcos
dx
x21
1
Cx sin
Cx 1sin
Integral Formulas to Memorize
• The same as all of the derivative formulas that are memorized.
• List on pg. 357 (and inside front cover of textbook).
More Difficult Examples
1) Find
2) Find
xdxcos5
dxxx 2
More Difficult Examples
1) Find
2) Find
xdxcos5
dxxx 2
xdxcos5
More Difficult Examples
1) Find
2) Find
xdxcos5
dxxx 2
xdxcos5 Cx )sin(5
More Difficult Examples
1) Find
2) Find
xdxcos5
dxxx 2
xdxcos5 Cx )sin(5 Cx sin5
More Difficult Examples
1) Find
2) Find
xdxcos5
dxxx 2
xdxcos5 Cx )sin(5 Cx sin5
Cxx
32
32
More Examples (S)
3) Find
4) Find
dxxxx 1723 26
dxx
xx
4
42 2
More Examples (S)
3) Find
4) Find
dxxxx 1723 26
dxx
xx
4
42 2
Cxxxx 237
2
7
3
2
7
3
More Examples
3) Find
4) Find
dxxxx 1723 26
dxx
xx
4
42 2
Cxxxx 237
2
7
3
2
7
3
dxx )2( 2
More Examples
3) Find
4) Find
dxxxx 1723 26
dxx
xx
4
42 2
Cxxxx 237
2
7
3
2
7
3
dxx )2( 2
Cxx
21
1
More Examples
3) Find
4) Find
dxxxx 1723 26
dxx
xx
4
42 2
Cxxxx 237
2
7
3
2
7
3
dxx )2( 2
Cxx
21
1
Cxx
21
Last Example5) Find dx
x
x 2sin
cos
Last Example5) Find dx
x
x 2sin
cosdx
xx
x )
sin
1(
sin
cos
Last Example5) Find dx
x
x 2sin
cosdx
xx
x )
sin
1(
sin
cos
dxxx )(csccot
Last Example5) Find dx
x
x 2sin
cosdx
xx
x )
sin
1(
sin
cos
dxxx )(csccot
Cx csc
Homework:
page 363
# 9 – 33 odd
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