Transcript AP Calculus Mrs. Mongold
AP Calculus Mrs. Mongold
The Fundamental Theorem of Calculus, Part 1 If
f
a x
dF dx
d dx
a x
First Fundamental Theorem:
d dx
a x
1. Derivative of an integral.
First Fundamental Theorem:
d d
x
a x
1. Derivative of an integral.
2. Derivative matches upper limit of integration.
First Fundamental Theorem:
d dx
a x
1. Derivative of an integral.
2. Derivative matches upper limit of integration.
3. Lower limit of integration is a constant.
First Fundamental Theorem:
d dx
a x
New variable.
1. Derivative of an integral.
2. Derivative matches upper limit of integration.
3. Lower limit of integration is a constant.
The long way: First Fundamental Theorem:
d
x
cos
x
1. Derivative of an integral.
dx d d dx dx
sin sin 2. Derivative matches
t x x
upper limit of integration.
sin 0 3. Lower limit of integration is a constant.
d dx
sin
x
cos
x
d dx
0
x
1 1+t 2
dt
1 1
x
2 1. Derivative of an integral.
2. Derivative matches upper limit of integration.
3. Lower limit of integration is a constant.
d dx
0
x
2 cos cos
d dx x
2 2
x
The upper limit of integration does not match the derivative, but we could use the
chain rule
.
d dx
x
5
d dx
5
x
3 sin
x
The lower limit of integration is not a constant, but the upper limit is. We can
change the sign
of the integral and
reverse the limits
.
d dx
2
x x
2 1 2
e t dt
Neither limit of integration is a constant.
We
split the integral into two parts
.
d dx
0
x
2 2 1
e t dt
2 0
x
1 2
e t dt
It does not matter what constant we use!
d dx
0
x
2 1 2
e t dt
0 2
x
1 2
e t dt
1 2
e x
2
2
x
1 2
e
2
x
2
2
x
2
e
(Limits are reversed.)
2 2
e
The Fundamental Theorem of Calculus, Part 2 If
f
F
is any antiderivative of
f
a b
(Also called the
Integral Evaluation Theorem
) We already know this!
To evaluate an integral, take the anti-derivatives and subtract.