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Area between curves Area between curves

Area between curves - PowerPoint Presentation

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Area between curves - PPT Presentation

Section 72a Area Between Curves Suppose we want to know the area of a region that is bounded above by one curve y f x and below by another y g x a b Upper curve ID: 613886

region area guided find area region find guided graph practice curves enclosed curve line boundedabove squared solve units lies andfrom negative axis

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Slide1

Area between curves

Section 7.2aSlide2

Area Between Curves

Suppose we want to know the area of a region that is bounded

above by one curve,

y = f(x), and below by another, y = g(x):

a

b

Upper curve

Lower curve

First, we partition the region

into vertical strips of equal

width and approximate

each strip as a rectangle

with area

Note: This expression will be

non-negative even if the region

lies below the

x

-axis.Slide3

Area Between Curves

Suppose we want to know the area of a region that is bounded

above by one curve,

y = f(x), and below by another, y = g(x):

a

b

Upper curve

Lower curve

We can approximate the

area of the region with the

Riemann sum

The limit of these sums as

isSlide4

Definition: Area Between Curves

If

f

and g are continuous with throughout[a, b], then the area between the curves y = f(

x) andy

= g(

x) from a

to b is the integral of [

f – g] from

a to b

,Slide5

Guided Practice

Find the area of the region between and

from to .

First, graph the two curves

over the given interval:

Now, use our new formula to

find the enclosed area:Slide6

Guided Practice

Find the area of the region between and

from to .

First, graph the two curves

over the given interval:Slide7

Guided Practice

Find the area of the region enclosed by the parabola

and the line .

The graph:

To find our limits of integration

(

a

and

b

), we need to solve

the system

a

b

Algebraically, or by calculator:Slide8

Guided Practice

Find the area of the region enclosed by the parabola

and the line .

The graph:

Because the parabola lies above

the line, we have

a

b

units

squaredSlide9

Guided Practice

Find the area of the region enclosed by the graphs of

and .

The graph:

a

b

To find our limits of integration

(

a

and

b

), we need to solve

the system

Solve graphically:

Store the negative value as A

and the positive value as B.Slide10

Guided Practice

Find the area of the region enclosed by the graphs of

and .

The graph:

a

b

Note: The trigonometric function

lies above the parabola…

units squared

Let’s evaluate this one numerically…

Area:Slide11

Guided Practice

Find the area of the region

R

in the first quadrant that is boundedabove by and below by the x-axis and the line .

The graph of

R

:

(4,2)

1

2

3

4

1

2

A

B

Area of region A:Slide12

Guided Practice

The graph of

R

:

(4,2)

1

2

3

4

1

2

A

B

Area of region B:Slide13

Guided Practice

The graph of

R

:

(4,2)

1

2

3

4

1

2

A

B

Area of

R

= Area of A + Area of B:

Units

squaredSlide14

Guided Practice

Find the area of the region enclosed by the given curves.

First, graph in [–2,12] by [0,3.5]

Intersection points:

Break into three

subregions

:Slide15

Guided Practice

Find the area of the region enclosed by the given curves.

First, graph in [–2,12] by [0,3.5]

Intersection points:

Break into three

subregions

: