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The Area Between Two Curves. Lesson 6.1. When f(x) < 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.

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when f x 0
When f(x) < 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
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
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
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) > 0 and f(x) < 0
    • Where f(x) < 0, take ______________ of the definite integral
try this
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) < 0
area between two curves
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
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
Slicing the Shark the Other Way
  • We could make these graphs as ________________
  • Now each slice is_______ by (k(y) – j(y))
practice
Practice
  • Determine the region bounded between the given curves
  • Find the area of the region
horizontal slices
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
Assignments A
  • Lesson 7.1A
  • Page 452
  • Exercises 1 – 45 EOO
integration as an accumulation process
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 process14
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
Try It Out
  • Find the accumulation function for
  • Evaluate
    • F(0)
    • F(4)
    • F(6)
applications
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
Assignments B
  • Lesson 7.1B
  • Page 453
  • Exercises 57 – 65 odd, 85, 88
revolving a function
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

dx

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?
disks21
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 out22
Try It Out!
  • Try the function y = x3 on the interval 0 < x < 2 rotated about x-axis
revolve about line not a coordinate 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
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
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
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 axis27
Revolving About y-Axis
  • Must consider curve asx = f(y)
    • Radius ____________
    • Slice is dy thick
  • Volume of the solid rotatedabout y-axis
flat washer
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 < y < 16
cross sections
Cross Sections
  • Consider a square at x = c with side equal to side s = f(c)
  • Now let this be a thinslab with thickness Δx
  • What is the volume of the slab?
  • Now sum the volumes of all such slabs

f(x)

c

a

b

cross sections30
Cross Sections
  • This suggests a limitand an integral

f(x)

c

a

b

cross sections31
Cross Sections
  • We could do similar summations (integrals) for other shapes
    • Triangles
    • Semi-circles
    • Trapezoids

f(x)

c

a

b

try it out32
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
Assignment
  • Lesson 7.2A
  • Page 463
  • Exercises 1 – 29 odd
  • Lesson 7.2B
  • Page 464
  • Exercises 31 - 39 odd, 49, 53, 57
slide35

Find the volume generated when this shape is revolved about the y axis.

We can’t solve for x, so we can’t use a horizontal slice directly.

slide36

If we take a ____________slice

and revolve it about the y-axis

we get a cylinder.

shell method
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
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 out39
Try It Out!
  • Consider the region bounded by x = 0, y = 0, and
hints for shell method
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
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
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 axis43
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 < x < a

try it out44
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 out45
Try It Out
  • Integral for the volume is
assignment46
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

Why?

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 length49
Arc Length
  • We sum the individual lengths
  • When we take a limit of the above, we get the integral
arc length50
Arc Length
  • Find the length of the arc of the function for 1 < x < 2
surface area of a cone

s

h

r

Surface Area of a Cone
  • Slant area of a cone
  • Slant area of frustum

L

surface area
Surface Area

Δx

  • Suppose we rotate thef(x) from slide 2 aroundthe x-axis
    • A surface is formed
    • A slice gives a __________

P1

Pi

Pn

P0

xi

b

a

Δs

surface area53
Surface Area
  • We add the cone frustum areas of all the slices
    • From a to b
    • Over entire length of the curve
surface area54
Surface Area
  • Consider the surface generated by the curve y2 = 4x for 0 < x < 8 about the x-axis
surface area55
Surface Area
  • Surface area =
limitations
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
assignment57
Assignment
  • Lesson 7.4
  • Page 383
  • Exercises 1 – 29 odd also 37 and 55,
slide58

Work

Lesson 7.5

slide59

50

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

a

b

x

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 law61
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 law62
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
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
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

r

b

a

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 liquids66
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
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 pistonΔx feet is
work done by expanding gas68
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 gas69
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
Assignment A
  • Lesson 7.5
  • Page 405
  • Exercises 1 – 41 EOO
slide72
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
slide73
Mass
  • The relationship is
  • Contrast of measures of mass and force
centroid
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

centroid75
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

centroid76

Also notated Mx,moment about x-axis

Centroid
  • Centroid for multiple points
  • Centroid about x-axis

First moment of the system

Also notated My, moment about y-axis

centroid77
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?
centroid78
Centroid
  • Given 10g at (2,-1), 7g at (4, 3), and 12g at (-5,2)

7g

12g

10g

centroid79

a

b

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, ρand the area
    • We sum the products of distance and mass

centroid of area under a curve
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
Centroid of Region Between Curves

f(x)

  • Moments
  • Mass

g(x)

Centroid

try it out82
Try It Out!
  • Find the centroid of the plane region bounded by y = x2 + 16 and the x-axis over the interval 0 < x < 4
    • Mx = ?
    • My = ?
    • m = ?
theorem of pappus
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

R

L

r

c

theorem of pappus84
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

R

L

r

c

assignment85
Assignment
  • Lesson 7.6
  • Page 504
  • Exercises 1 – 41 EOO also 49
fluid pressure
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 pressure88
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
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 pressure90
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 pressure91

2.5 - y

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 a92
Assignment A
  • Lesson 7.7
  • Page 511
  • Exercises 1-25 odd