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6034 Fundamental Theorem of Calculus (Part 2)
AB CALCULUS
The Indefinite Integral (Antiderivative) finds a Family of Functions whose derivative is given.
( ) cos( )A x t dt
Given an Initial Condition we find the Particular Function
( ) 32
f
𝐴 (𝑥 )=sin (𝑡 )+𝑐
3=sin ( 𝜋2 )+𝑐3=1+𝑐
2=𝑐𝐴 (𝑥 )=sin( 𝜋2 )+2
The Definite Integral as a Particular Function:
0
( ) cos( )x
A x t dtEvaluate at 0, , ,
6 4 3x
Evaluate the Definite Integral for each of these points.
The Definite Integral is actually finding points on the Accumulation graph.
Evaluate the definite integral.
¿ sin (𝑡)|𝑥0¿ sin (𝑥 )− sin(0)
𝐴 (𝑥 )=sin(𝑥)
0
0
cos (𝑡 ) 𝑑𝑡=0
0
𝜋6
cos (𝑡 ) 𝑑𝑡=12
0
𝜋4
cos (𝑡 ) 𝑑𝑡=√22
0
𝜋3
cos (𝑡 ) 𝑑𝑡=√32
1
Since A(x) is a function, what then is the rate of change of that function?
0
( ) cos( )x
A x t dt( ) sin( )A x x
( ) cos( )A x x
In words, integration and differentiation are inverse operations
Take derivative
2nd Fundamental Theorem of Calculus
Given: , we want to find
Note: a is a constant, u is a function of x; and the order matters!
( ) ( )x
a
A x f t dt/ ( )A x
( ) ( )u
a
df t dt f u u
dx
2nd Fundamental Theorem of Calculus: If f is continuous on an open interval, I, containing a point, a,
then for every x in I :
“a” is a constant
Demonstration: < function x only >
find
2
( ) sin( )x
A x t dt
( ( ))dA x
dx
2
( ( )) sin( ) ]xd d
A x t dtdx dx
In Words:Sub in the function u and multiply by derivative of u
−sin (𝑥 )∗1
Example:
Find and verify:
2
1
1xdt dt
dx
2 1x
this
Not this
(𝑥2+1 ) (1 ) 𝑡3
3+𝑡|𝑥1
=
𝐴′ (𝑥 )=(𝑥2+1 )
−( 13+1)=− 4
3
Example:
Find without Integrating:
2
3
1xdt dt
dx
√𝑥2+1 (1 )
THE COMPOSITE FUNCTION
If g(x) is given instead of x:
In words: Substitute in g(x) for t and then multiply by the derivative of g(x)…exactly the chain rule
(derivative of the outside * derivative of the inside)
( )/ ( ( )) ( )
g x
a
dQ g x f t dt
dx
( )[ ( )]g xadF t
dx
( ( )) ( )dF g x F a
dx
( ( ))* '( ) ( ( ))* '( )F g x g x or f g x g x
THE COMPOSITE FUNCTION
If , (a composite function)
then
( )u g x
/( ) ( )*u
a
df t dt f u u
dx
In Words:Sub u in for t and multiply by u’
Demonstration: < The composite function >
Find:
In Words:
3
4
cos( )xd
t dtdx
3 2cos( )*(3 )x x
¿cos (𝑥3 ) 3𝑥2
Verify 𝑦=sin 𝑥3− sin𝜋4
=sin 𝑥3− √22
𝑦 ′=cos 𝑥3 (3 𝑥2)Sub in for t and multiply by the derivative of
Example :
Find without Integrating:
If , solve for
2
22
1( )
x
Q x dtt
/ ( )Q x
1
(𝑥2)2(2𝑥 )=2 𝑥
𝑥4 =2
𝑥3
Example: Rewriting the Integral
2
5
(2 5)x
t dt
Find without integrating: Show middle stepdy
dx
−5
𝑥2
(2𝑡−5 )𝑑𝑡
𝑦 ′=− ( 2𝑥2−5 ) (2𝑥 )
Example: Rewriting the Integral - Two variable limits:
Find without Integrating:
break into two parts . . . . . chose any number in domain of for a and rewrite into required form
.
sin( )
cos( )
1x
x
dt dt
dx
1t
c os𝑥
0
√𝑡+1𝑑𝑡+ 0
sin 𝑥
√𝑡+1𝑑𝑡
− 0
cos 𝑥
√𝑡+1𝑑𝑡+ 0
sin 𝑥
√𝑡+1𝑑𝑡
−√cos𝑥+1 (−sin 𝑥 )+√sin 𝑥+1 (cos𝑥 )
Last Update:
• 1/25/11
• Worksheet