5.4 Fundamental Theorem of Calculus Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 1998 Morro Rock, California
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.