5.4 fundamental theorem of calculus greg kelly, hanford high school, richland, washingtonphoto by...
TRANSCRIPT
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
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.
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
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.
First 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:
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.
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.
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.
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.
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.