The Area Between Two Curves

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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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## The Area Between Two Curves

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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
• 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) > 0 and f(x) < 0
• Where f(x) < 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) < 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

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 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 < x < 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?

• 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?
• Must consider curve asx = f(y)
• 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
• f(y) = _____
• Limits?
• 0 < y < 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 Δx
• What is the volume of the slab?
• Now sum the volumes of all such slabs

f(x)

c

a

b

Cross Sections
• This suggests a limitand an integral

f(x)

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

We can’t solve for x, so we can’t 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
• Rotate the region bounded by y = 4x and y = x2 about the x-axis
• What are the dimensions needed?
• height
• thickness

thickness = _____

_______________ = y

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

g(x)

f(x)

x = a

• 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 Out
• Rotate the region bounded by 4 – x2 , x = 0 and, y = 0 about the line x = 2
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

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 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 < x < 2

s

h

r

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

L

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 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 < x < 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

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

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 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 (__________________)

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 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 pistonΔx 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

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

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
• 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 < x < 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

R

L

r

c

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

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

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