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4.4 The Fundamental Theorem of
Calculus
Essential Question: How are the integral & the
derivative related?
The Fundamental Thm, Part 1:
where F = f.
If f is continuous on [a, b], then
)()( aFbFdxxfb
a
Ex 1: Find the area under the
parabola y = x2 from 0 to 1.
Ex 2: Evaluate the integral.
dxex3
1
Ex 3: Evaluate the integral.
6
3
x
dx
The Fundamental Thm, Part 2:
bxa
dttfxgx
a
IF
The Fundamental Thm, Part 2:
AND f is continuous on [a, b],
THEN g is continuous on [a, b] AND differentiable
on (a, b), AND …
The Fundamental Thm, Part 2:
g (x) = f (x)
WOW!
Ex 4: Sketch the graph of g.
dttxgx
0
2
Ex 5: Find the derivative of g.
dttxgx
0
21
Ex 6: Find
dttdx
d x sec
4
0
HW: 4.4 pg. 291 # 5 – 25 EOO, #27 – 37 odds, #51, 55 – 59 all,
#67 – 93 odds, 74