4.4 The Fundamental Theorem of Calculus. Essential Question: How are the integral & the derivative related?

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