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5.4 Fundamental Theorem of Calculus
Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 1998
Morro Rock, California
If you were being sent to a desert island and could take only one equation with you,
x
a
d f t dt f xdx
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.
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 d f t dt f xdx dx
x
a
d f t dt f xdx
First Fundamental Theorem:
1. Derivative of an integral.
a
xd f t dtx
f xd
2. Derivative matches upper limit of integration.
First Fundamental Theorem:
1. Derivative of an integral.
a
xd f t dt f xdx
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
d f t dt f xdx
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:
cos xd t dt
dx cos x 1. Derivative of an integral.
2. Derivative matches upper limit of integration.
3. Lower limit of integration is a constant.
sin xd tdx
sin sind xdx
0
sind xdx
cos x
The long way: First Fundamental Theorem:
20
1 1+txd dt
dx 2
11 x
1. Derivative of an integral.
2. Derivative matches upper limit of integration.
3. Lower limit of integration is a constant.
2
0cos
xd t dtdx
2 2cos dx xdx
2cos 2x x
22 cosx x
The upper limit of integration does not match the derivative, but we could use the chain rule.
53 sin
x
d t t dtdx
The lower limit of integration is not a constant, but the upper limit is.
53 sin xd t t dt
dx
3 sinx x
We can change the sign of the integral and reverse the limits.
2
2
1 2
x
tx
d dtdx e
Neither limit of integration is a constant.
2 0
0 2
1 1 2 2
x
t tx
d dt dtdx e e
It does not matter what constant we use!
2 2
0 0
1 1 2 2
x x
t t
d dt dtdx e e
2 2
1 12 222 xx
xee
(Limits are reversed.)
(Chain rule is used.)2 2
2 222 xx
xee
We split the integral into two parts.
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.