Sections 4.2-4.4: Area, Definite Integrals, and The Fundamental Theorem of Calculus

Sections 4.2-4.4: Area, Definite Integrals, and The Fundamental

Theorem of Calculus

Distance TraveledA train moves along a track at a steady rate of 75 miles per

hour from 3:00 A.M. to 8:00 A.M. What is the total distance traveled by the train?

Time (h)

Velo

city

(mph

) Applying the well known formula:

distance rate time 75mph 5hr 375 miles

Notice the distance traveled by the train (375 miles) is exactly the area of the

rectangle whose base is the time interval [3,8] and whose height is the constant

velocity function v=75.

Distance TraveledA particle moves along the x-axis with velocity v(t)=-t2+2t+5

for time t≥0 seconds. How far is the particle after 3 seconds?

Time

Velo

city

The distance traveled is still the area under the curve.

Unfortunately the shape is a irregular region. We need to

find a method to find this area.

The Area ProblemWe now investigate how to solve the area

problem: Find the area of the region S that lies under the curve y=f(x) from a to b.

a b

f(x)

S

This means S is bounded by the graph of a continuous function, two vertical lines, and

the x-axis.

Finding AreaIt is easy to calculate the area of certain

shapes because familiar formulas exist:

A=lw A=½bhThe area of irregular polygons can be found

by dividing them into convenient shapes and their areas:

A1

A2 A

3 A4

1 2 3 4A A A A A

Approximating the Area Under a Curve

a b

We first approximate the area under a function by rectangles.


a b

Then we take the limit of the areas of these rectangles as we increase the number of rectangles.


a b

Then we take the limit of the areas of these rectangles as we increase the number of rectangles.

Estimating Area Using Rectangles and Right Endpoints

Use rectangles to estimate the area under the parabola y=x2 from 0 to 1 using 4 rectangles and right endpoints.

12

14

34

Divide the area under the curve into 4 equal strips

Make rectangles whose base is the same as the strip and whose height is the same as

the right edge of the strip.

Find the Sum of the Areas:

1 14 4A f 1 1

4 2f 314 4f 1

4 1f

21 14 4 21 1

4 2 2314 4 21

4 1

0.46875

Width = ¼ and height = value of the function

at ¼

Estimating Area Using Rectangles and Right Endpoints

Use rectangles to estimate the area under the parabola y=x2 from 0 to 1 using 8 rectangles and right endpoints.

12

14

34



the right edge of the strip.


1 18 8A f 1 1

8 4f 318 8f 1 1

8 2f

0.3984375

Width = 1/8 and height = value of the

function at 1/8

38

58

78

18

518 8f 31

8 4f 718 8f 1

8 1f

21 18 8 21 1

8 4 2318 8 21 1

8 2

2518 8 231

8 4 2718 8 21

8 1

Estimating Area Using Rectangles and Left Endpoints

Use rectangles to estimate the area under the parabola y=x2 from 0 to 1 using 4 rectangles and left endpoints.

12

14

34



the left edge of the strip.


14 0A f 1 1

4 4f 1 14 2f 31

4 4f

214 0 21 1

4 4 21 14 2 231

4 4

0.21875

Width = ¼ and height = value of the function

at 0

Estimating Area Using Rectangles and Left Endpoints

Use rectangles to estimate the area under the parabola y=x2 from 0 to 1 using 8 rectangles and left endpoints.

12

14

34



the left edge of the strip.


18 0A f 1 1

8 8f 1 18 4f 31

8 8f

0.2734375

Width = 1/8 and height = value of the

function at 0

38

58

78

18

1 18 2f 51

8 8f 318 4f 71

8 8f

218 0 21 1

8 8 21 18 4 231

8 8

21 18 2 251

8 8 2318 4 271

8 8

Distance TraveledA particle moves along the x-axis with velocity v(t)=-t2+2t+5

for time t≥0 seconds. Use three midpoint rectangles to estimate how far the particle traveled after 3 seconds?

Time

Velo

city



the middle of the strip.


1 0.5D v 1 1.5v 1 2.5v

21 0.5 2 0.5 5

21 1.5 2 1.5 5

21 2.5 2 2.5 5

15.25

Width = 1 and height = value of the function at

0.5

Negative AreaIf a function is less than zero for an interval,

the region between the graph and the x-axis represents negative area.

Positive Area

Negative Area

Definite Integral: Area Under a Curve

If y=f(x) is integrable over a closed interval [a,b], then the area under the curve y=f(x) from a to b is the integral of f from a to b.

area above the -axis area below the -axisb

af x dx x x

Upper limit of integration

Lower limit of integration

The Existence of Definite IntegralsAll continuous functions are integrable. That is, if a function

f is continuous on an interval [a,b], then its definite integral over [a,b] exists .

Ex: 9

14 x dx 1

2 3 3 12 5 5

8

Rules for Definite Integrals

Constant Multiple

Sum Rule

Difference Rule

Additivity

a a

b bkf x dx k f x dx

a a a

b b bf x g x dx f x dx g x dx

a a a

b b bf x g x dx f x dx g x dx

Let f and g be functions and x a variable; a, b, c, and k be constant.

b c c

a b af x dx f x dx f x dx

The First Fundamental Theorem of Calculus

If f is continuous on the interval [a,b] and F is any function that satisfies F '(x) = f(x) throughout this interval then

b

af x dx F b F a

b

av t dt s b s a

Example 1Evaluate

First Find the indefinte integral

F(x): 2F x x dx 2 11

2 1 x C 31

3 x C

13

Now apply the FTC to find the definite

integral:

F b F a 1 0F F

3 31 13 31 0C C

13 C C Notice that it is

not necessary to include the “C”

with definite integrals

1 2

0x dx

2

2cos x dx

Examples: New Notation1. Evaluate

2

2sin x

2 2sin sin 1 1 F(x) Bounds

2

5

310 dx

2. Evaluate

5

310x

10 5 10 3 50 30 80

91

4 xx dx3. Evaluate

91 2 212 4

2x x 1 2 2 1 2 21 12 22 9 9 2 4 4

34.5

9 1 2

4x x dx

Sections 4.2-4.4: Area, Definite Integrals, and The Fundamental Theorem of Calculus

Documents

estimating area

area problemwe

number of rectangles

left endpointsuse rectangles

right endpointsuse rectangles

area of certain shapes

equal stripsmake rectangles

curve y