fundamental theorems of calculus 6.4. the first (second?) fundamental theorem of calculus if f is...
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Fundamental Theorems of Calculus
6.4
The First (second?) Fundamental Theorem of Calculus
If f is continuous on , then the function ,a b
x
aF x f t dt
has a derivative at every point in , and ,a b
x
a
dF df t dt f x
dx dx
x
a
df t dt f x
dx
First Fundamental Theorem:
1. Derivative of an integral.
a
xdf t dt
xf x
d
2. Derivative matches upper limit of integration.
First Fundamental Theorem:
1. Derivative of an integral.
a
xdf t dt f x
dx
1. Derivative of an integral.
2. Derivative matches upper limit of integration.
3. Lower limit of integration is a constant.
First Fundamental Theorem:
x
a
df t dt f x
dx
1. Derivative of an integral.
2. Derivative matches upper limit of integration.
3. Lower limit of integration is a constant.
New variable.
Second Fundamental Theorem:
cos xd
t dtdx cos x 1. Derivative of an integral.
2. Derivative matches upper limit of integration.
3. Lower limit of integration is a constant.
sinxdt
dx
sin sind
xdx
0
sind
xdx
cos x
The long way:First Fundamental Theorem:
Example Applying the Fundamental Theorem
Find sin .xdtdt
dx
sin sinxdtdt x
dx
20
1
1+t
xddt
dx 2
1
1 x
1. Derivative of an integral.
2. Derivative matches upper limit of integration.
3. Lower limit of integration is a constant.
Example Variable Lower Limits of Integration
5Find if sin .x
dyy t tdt
dx
2
1cos
xdtty
2
2
1
2
1
cos2
2)cos(
cos
cos
cos
xx
xx
dx
duu
dx
dudtt
du
ddx
du
du
dy
dx
dy
xuanddtty
u
u
Example The Fundamental Theorem with the Chain Rule
2
1Find / if sin .xdy dx y tdt
2
2
1
2
x
tx
ddt
dx eNeither limit of integration is a constant.
2 0
0 2
1 1
2 2
x
t tx
ddt dt
dx e e
It does not matter what constant we use!
2 2
0 0
1 1
2 2
x x
t t
ddt dt
dx e e
2 2
1 12 2
22xx
xee
(Limits are reversed.)
(Chain rule is used.)2 2
2 2
22xx
x
ee
We split the integral into two parts.
Second (first?) FTOC
b
aaFbFdxxf )()()(
3
1
3 )1( dxx
24
14
13
4
81
4
3
1
4
xx
)(0)(
))()(()(
)()(
)()(
xfxf
aFxFDdttfD
aFxF
tFdttf
x
a xx
x
a
x
a
How to Find Total Area Analytically
To find the area between the graph of ( ) and the -axis over the interval
[ , ] analytically,
1. partition [ , ] with the zeros of ,
2. integrate over each subinterval,
3. add the absolute values o
y f x x
a b
a b f
f
f the integrals.
How to Find Total Area Analytically