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The Area Between Two Curves

- Lesson 6.1

When f(x) lt 0

- Consider taking the definite integral for the

function shown below. - The integral gives a ___________ area
- We need to think of this in a different way

a

b

f(x)

Another Problem

- What about the area between the curve and the

x-axis for y x3 - What do you get forthe integral?
- Since this makes no sense we need another way

to look at it

Solution

- We can use one of the properties of integrals
- We will integrate separately for _________ and

__________

We take the absolute value for the interval which

would give us a negative area.

General Solution

- When determining the area between a function and

the x-axis - Graph the function first
- Note the ___________of the function
- Split the function into portions where f(x) gt 0

and f(x) lt 0 - Where f(x) lt 0, take ______________ of the

definite integral

Try This!

- Find the area between the function h(x)x2 x

6 and the x-axis - Note that we are not given the limits of

integration - We must determine ________to find limits
- Also must take absolutevalue of the integral

sincespecified interval has f(x) lt 0

Area Between Two Curves

- Consider the region betweenf(x) x2 4 and

g(x) 8 2x2 - Must graph to determine limits
- Now consider function insideintegral
- Height of a slice is _____________
- So the integral is

The Area of a Shark Fin

- Consider the region enclosed by
- Again, we must split the region into two parts
- _________________ and ______________

Slicing the Shark the Other Way

- We could make these graphs as ________________
- Now each slice is_______ by (k(y) j(y))

Practice

- Determine the region bounded between the given

curves - Find the area of the region

Horizontal Slices

- Given these two equations, determine the area of

the region bounded by the two curves - Note they are x in terms of y

Assignments A

- Lesson 7.1A
- Page 452
- Exercises 1 45 EOO

Integration as an Accumulation Process

- Consider the area under the curve y sin x
- Think of integrating as an accumulation of the

areas of the rectangles from 0 to b

b

Integration as an Accumulation Process

- We can think of this as a function of b
- This gives us the accumulated area under the

curve on the interval 0, b

Try It Out

- Find the accumulation function for
- Evaluate
- F(0)
- F(4)
- F(6)

Applications

- The surface of a machine part is the region

between the graphs of y1 x and y2 0.08x2

k - Determine the value for k if the two functions

are tangent to one another - Find the area of the surface of the machine part

Assignments B

- Lesson 7.1B
- Page 453
- Exercises 57 65 odd, 85, 88

Volumes The Disk Method

- Lesson 7.2

Revolving a Function

- Consider a function f(x) on the interval a, b
- Now consider revolvingthat segment of curve

about the x axis - What kind of functions generated these solids of

revolution?

f(x)

a

b

Disks

f(x)

- We seek ways of usingintegrals to determine

thevolume of these solids - Consider a disk which is a slice of the solid
- What is the radius
- What is the thickness
- What then, is its volume?

Disks

- To find the volume of the whole solid we sum

thevolumes of the disks - Shown as a definite integral

f(x)

a

b

Try It Out!

- Try the function y x3 on the interval 0 lt x lt

2 rotated about x-axis

Revolve About Line Not a Coordinate Axis

- Consider the function y 2x2 and the boundary

lines y 0, x 2 - Revolve this region about the line x 2
- We need an expression forthe radius____________

___

Washers

- Consider the area between two functions rotated

about the axis - Now we have a hollow solid
- We will sum the volumes of washers
- As an integral

f(x)

g(x)

a

b

Application

- Given two functions y x2, and y x3
- Revolve region between about x-axis

What will be the limits of integration?

Revolving About y-Axis

- Also possible to revolve a function about the

y-axis - Make a disk or a washer to be ______________
- Consider revolving a parabola about the y-axis
- How to represent the radius?
- What is the thicknessof the disk?

Revolving About y-Axis

- Must consider curve asx f(y)
- Radius ____________
- Slice is dy thick
- Volume of the solid rotatedabout y-axis

Flat Washer

- Determine the volume of the solid generated by

the region between y x2 and y 4x, revolved

about the y-axis - Radius of inner circle?
- f(y) _____
- Radius of outer circle?
- Limits?
- 0 lt y lt 16

Cross Sections

- Consider a square at x c with side equal to

side s f(c) - Now let this be a thinslab with thickness ?x
- What is the volume of the slab?
- Now sum the volumes of all such slabs

f(x)

c

a

b

Cross Sections

f(x)

- This suggests a limitand an integral

c

a

b

Cross Sections

- We could do similar summations (integrals) for

other shapes - Triangles
- Semi-circles
- Trapezoids

f(x)

c

a

b

Try It Out

- Consider the region bounded
- above by y cos x
- below by y sin x
- on the left by the y-axis
- Now let there be slices of equilateral triangles

erected on each cross section perpendicular to

the x-axis - Find the volume

Assignment

- Lesson 7.2A
- Page 463
- Exercises 1 29 odd
- Lesson 7.2B
- Page 464
- Exercises 31 - 39 odd, 49, 53, 57

Volume The Shell Method

- Lesson 7.3

Find the volume generated when this shape is

revolved about the y axis.

We cant solve for x, so we cant use a

horizontal slice directly.

If we take a ____________slice

and revolve it about the y-axis

we get a cylinder.

Shell Method

- Based on finding volume of cylindrical shells
- Add these volumes to get the total volume
- Dimensions of the shell
- _________of the shell
- _________of the shell
- ________________

The Shell

- Consider the shell as one of many of a solid of

revolution - The volume of the solid made of the sum of the

shells

dx

f(x)

f(x) g(x)

x

g(x)

Try It Out!

- Consider the region bounded by x 0, y 0, and

Hints for Shell Method

- Sketch the __________over the limits of

integration - Draw a typical __________parallel to the axis of

revolution - Determine radius, height, thickness of shell
- Volume of typical shell
- Use integration formula

Rotation About x-Axis

- Rotate the region bounded by y 4x and y x2

about the x-axis - What are the dimensions needed?
- radius
- height
- thickness

thickness _____

_______________ y

Rotation About Non-coordinate Axis

- Possible to rotate a region around any line
- Rely on the basic concept behind the shell method

g(x)

f(x)

x a

Rotation About Non-coordinate Axis

- What is the radius?
- What is the height?
- What are the limits?
- The integral

r

g(x)

f(x)

a x

x c

x a

f(x) g(x)

c lt x lt a

Try It Out

- Rotate the region bounded by 4 x2 , x 0 and,

y 0 about the line x 2 - Determine radius, height, limits

Try It Out

- Integral for the volume is

Assignment

- Lesson 7.3
- Page 472
- Exercises 1 25 odd
- Lesson 7.3B
- Page 472
- Exercises 27, 29, 35, 37, 41, 43, 55

Arc Length and Surfaces of Revolution

- Lesson 7.4

Arc Length

- We seek the distance along the curve fromf(a)

to f(b) - That is from P0 to Pn
- The distance formula for each pair of points

P1

Pi

Pn

P0

b

a

What is another way of representing this?

Arc Length

- We sum the individual lengths
- When we take a limit of the above, we get the

integral

Arc Length

- Find the length of the arc of the function for

1 lt x lt 2

Surface Area of a Cone

- Slant area of a cone
- Slant area of frustum

L

Surface Area

?x

- Suppose we rotate thef(x) from2

aroundthe x-axis - A surface is formed
- A slice gives a __________

P1

Pi

Pn

P0

xi

b

a

?s

Surface Area

- We add the cone frustum areas of all the slices
- From a to b
- Over entire length of the curve

Surface Area

- Consider the surface generated by the curve y2

4x for 0 lt x lt 8 about the x-axis

Surface Area

- Surface area

Limitations

- We are limited by what functions we can integrate
- Integration of the above expression is not

_________________________ - We will come back to applications of arc length

and surface area as new integration techniques

are learned

Assignment

- Lesson 7.4
- Page 383
- Exercises 1 29 odd also 37 and 55,

Work

- Lesson 7.5

Work

- DefinitionThe product of
- The ____________exerted on an object
- The _______________the object is moved by the

force - When a force of 50 lbs is exerted to move an

object 12 ft. - 600 ft. lbs. of work is done

12 ft

Hooke's Law

- Consider the work done to stretch a spring
- Force required is proportional to _________
- When k is constant of proportionality
- Force to move dist x
- Force required to move through i th interval,

?x - ?W F(xi) ?x

Hooke's Law

- We sum those values using the definite integral
- The work done by a ____________force F(x)
- Directed along the x-axis
- From x a to x b

Hooke's Law

- A spring is stretched 15 cm by a force of 4.5 N
- How much work is needed to stretch the spring 50

cm? - What is F(x) the force function?
- Work done?

Winding Cable

- Consider a cable being wound up by a winch
- Cable is 50 ft long
- 2 lb/ft
- How much work to wind in 20 ft?
- Think about winding in ?y amt
- y units from the top ? 50 y ft hanging
- dist ?y
- force required (weight) 2(50 y)

Pumping Liquids

- Consider the work needed to pump a liquid into or

out of a tank - Basic concept Work weight x _____________
- For each ?V of liquid
- Determine __________
- Determine dist moved
- Take summation (__________________)

Pumping Liquids Guidelines

- Draw a picture with thecoordinate system
- Determine _______of thinhorizontal slab of

liquid - Find expression for work needed to lift this slab

to its destination - Integrate expression from bottom of liquid to the

top

Pumping Liquids

4

- Suppose tank has
- r 4
- height 8
- filled with petroleum (54.8 lb/ft3)
- What is work done to pump oil over top
- Disk weight?
- Distance moved?
- Integral?

8

___________

Work Done by Expanding Gas

- Consider a piston of radius r in a cylindrical

casing as shown here - Let p pressure in lbs/ft2
- Let V volume of gas in ft3
- Then the work incrementinvolved in moving the

piston?x feet is

Work Done by Expanding Gas

- So the total work done is the summation of all

those increments as the gas expands from V0 to

V1 - Pressure is inversely proportionalto volume so

p _________ and

Work Done by Expanding Gas

- A quantity of gas with initial volume of1 cubic

foot and a pressure of 2500 lbs/ft2 expands to a

volume of 3 cubit feet. - How much work was done?

Assignment A

- Lesson 7.5
- Page 405
- Exercises 1 41 EOO

Moments, Center of Mass, Centroids

- Lesson 7.6

Mass

- Definition mass is a measure of a body's

____________to changes in motion - It is ___________ a particular gravitational

system - However, mass is sometimes equated with

__________ (which is not technically correct) - Weight is a type of ___________ dependent on

gravity

Mass

- The relationship is
- Contrast of measures of mass and force

Centroid

- Center of mass for a system
- The point where all the mass seems to be

concentrated - If the mass is of constant density this point is

called the __________________

4kg

10kg

6kg

Centroid

- Each mass in the system has a "moment"
- The product of ____________________________ from

the origin - "First moment" is the __________of all the

moments - The centroid is

4kg

10kg

6kg

Centroid

- Centroid for multiple points
- Centroid about x-axis

First moment of the system Also notated My,

moment about y-axis

Centroid

- The location of the centroid is the ordered pair
- Consider a system with 10g at (2,-1), 7g at (4,

3), and 12g at (-5,2) - What is the center of mass?

Centroid

- Given 10g at (2,-1), 7g at (4, 3), and 12g at

(-5,2)

7g

12g

10g

Centroid

- Consider a region under a curve of a material of

uniform density - We divide the region into ____________
- Mass of each considered to be centered at

_______________________center - Mass of each is the product of the density, ? and

the area - We sum the products of distance and mass

Centroid of Area Under a Curve

- First moment with respectto the y-axis
- First moment with respectto the x-axis
- Mass of the region

Centroid of Region Between Curves

f(x)

- Moments
- Mass

g(x)

Centroid

Try It Out!

- Find the centroid of the plane region bounded

by y x2 16 and the x-axis over the

interval 0 lt x lt 4 - Mx ?
- My ?
- m ?

Theorem of Pappus

- Given a region, R, in the plane and L a line in

the same plane and not intersecting R. - Let c be the centroid and r be the distance from

L to the centroid

L

Theorem of Pappus

- Now revolve the region about the line L
- Theorem states that the volume of the solid of

revolution iswhere A is the area of R

L

Assignment

- Lesson 7.6
- Page 504
- Exercises 1 41 EOO also 49

Fluid Pressure and Fluid Force

- Lesson 7.7

Fluid Pressure

- Definition The pressure on an object at depth h

is - Where w is the weight-density of the liquid per

unit of volume - Some example densitieswater 62.4

lbs/ft3mercury 849 lbs/ft3

Fluid Pressure

- Pascal's Principle pressure exerted by a fluid

at depth h is transmitted _______in all

__________________ - Fluid pressure given in terms of force per unit

area

Fluid Force on Submerged Object

- Consider a rectangular metal sheet measuring 2 x

4 feet that is submerged in 7 feet of water - Rememberso P 62.4 x 7 436.8
- And F P x Aso F 436.8 x 2 x 4 3494.4 lbs

Fluid Pressure

- Consider the force of fluidagainst the side

surface of the container - Pressure at a point
- Density x g x depth
- Force for a horizontal slice
- Density x g x depth x Area
- Total force

Fluid Pressure

- The tank has cross sectionof a trapazoid
- Filled to 2.5 ft with water
- Water is 62.4 lbs/ft3
- Function of edge
- Length of strip
- Depth of strip
- Integral

(-4,2.5)

(4,2.5)

(-2,0)

(2,0)

Assignment A

- Lesson 7.7
- Page 511
- Exercises 1-25 odd

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