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5.4 Fundamental Theorem of Calculus
Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 1998
Morro Rock, California
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If you were being sent to a desert island and could take only one equation with you,
x
a
df t dt f x
dx
might well be your choice.
Here is my favorite calculus textbook quote of all time, from CALCULUS by Ross L. Finney and George B. Thomas, Jr., ©1990.
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The Fundamental Theorem of Calculus, Part 1
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
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x
a
df t dt f x
dx
First Fundamental Theorem:
1. Derivative of an integral.
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a
xdf t dt
xf x
d
2. Derivative matches upper limit of integration.
First Fundamental Theorem:
1. Derivative of an integral.
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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:
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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.
First Fundamental Theorem:
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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:
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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.
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2
0cos
xdt dt
dx
2 2cosd
x xdx
2cos 2x x
22 cosx x
The upper limit of integration does not match the derivative, but we could use the chain rule.
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53 sin
x
dt t dt
dxThe lower limit of integration is not a constant, but the upper limit is.
53 sin xdt t dt
dx
3 sinx x
We can change the sign of the integral and reverse the limits.
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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.
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The Fundamental Theorem of Calculus, Part 2
If f is continuous at every point of , and if
F is any antiderivative of f on , then
,a b
b
af x dx F b F a
,a b
(Also called the Integral Evaluation Theorem)
We already know this!
To evaluate an integral, take the anti-derivatives and subtract.