THE FUNDAMENTAL THEOREM OF CALCULUS

THE FUNDAMENTAL THEOREM OF CALCULUS

Section 4.4Section 4.4

When you are done with your homework, you should

be able to…– Evaluate a definite integral using the

Fundamental Theorem of Calculus– Understand and use the Mean Value

Theorem for Integrals– Find the average value of a function

over a closed interval– Understand and use the Second

Fundamental Theorem of Calculus

•  

Galileo lived in Italy from 1570-1642. He defined science as the quantitative

description of nature—the study of time, distance and mass. He invented the 1st

accurate clock and telescope. Name one of his advances.

A. He discovered laws of motion for a falling object.

B. He defined science.C. He formulated the language of physics..D. All of the above.

THE FUNDAMENTAL THEOREM OF CALCULUS

• Informally, the theorem states that differentiation and definite integrals are inverse operations

• The slope of the tangent line was defined using the quotient

• The area of a region under a curve was defined using the product

– The Fundamental Theorem of Calculus states that the limit processes used to define the derivative and definite integral preserve this relationship

dy

dx

dydx

Secantl ine

y

x

Tangentl ine

y

x

Area ofRectangle

y

x

Slopey

x

Slope

y

x

Area y x Area y x

Area ofRegionundercurve

y

x

Theorem: The Fundamental Theorem of Calculus

• If a function f is continuous on the closed interval and F is an antiderivative of f on the interval , then

,a b ,a b

b

a

f x dx F b F a

Guidelines for Using the Fundamental Theorem of Calculus

1. Provided you can find an antiderivative of f, you now have a way to evaluate a definite integral without having to use the limit of a sum.

2. When applying the Fundamental Theorem of Calculus, the following notation is convenient:

3. It is not necessary to include a constant of integration in the antiderivative because

bb

aa

f x dx F x

F b F a

bb

aa

f x dx F x C

F b C F a C

F b F a

Example: Find the area of the region bounded by the graph

of , the x-axis, and the

vertical lines and .

2 3y x 1x 3x

15

10

5

-4 -2 2 4

r y = 3

q y = 1

h x = x2+3

Find the area under the curve bounded by the

graph of , , and the x-axis and the

y-axis.9/40.0

3 2f x x 1x

4

2

-5 5

h y = 0

g y = -1

f x = -x3+2

THE MEAN VALUE THEOREM FOR INTEGRALS

If f is continuous on the closed interval , then there exists a number c in the closed interval such that

  •   

,a b ,a b

b

a

f x dx f c b a

So…what does this mean?!Somewhere between the inscribed and

circumscribed rectangles there is a rectangle whose area is precisely equal to the area of the region under the curve.

Mean Value Rectangle

f c

ba c

AVERAGE VALUE OF A FUNCTION

• If f is integrable on the closed interval , then the average value of f on the interval is

,a b

1 b

a

f x dxb a

Find the average value of the function

13.00.0

32

1

3x dx

THE SECOND FUNDAMENTAL THEOREM OF CALCULUS

If f is continuous on an open interval I containing c, then, for every x in the interval,

x

a

df t dt f x

dx