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When f(x) < 0
• Consider taking the definite integral for the
function shown below.
The Area Between Two Curves
a
b
f(x)
Lesson 7.1
• The integral gives a ___________ area
• We need to think of this in a different way
Another Problem
• What about the area between the curve and the
x-axis for y = x3
• What do you get for
the integral?
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.
• Since this makes no sense – we need another way
to look at it
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 absolute
value of the integral since
specified interval has f(x) < 0
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Area Between Two Curves
• Consider the region between
f(x) = x2 – 4 and g(x) = 8 – 2x2
• Must graph to determine limits
The Area of a Shark Fin
• Consider the region enclosed by
• Now consider function inside
integral
• Height of a slice is _____________
• So the integral is
Slicing the Shark the Other Way
• We could make these graphs as ________________
• Again, we must split the region into two parts
• _________________ and ______________
Practice
• Determine the region bounded between the
given curves
• Find the area of the region
• Now each slice is
_______ by (k(y) – j(y))
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 454
• Exercises 1 – 45 EOO
61, 63
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Integration as an Accumulation Process
• Consider the area under the curve y = sin x
Integration as an Accumulation Process
• We can think of this as a function of b
b
• Think of integrating as an accumulation of the
areas of the rectangles from 0 to b
Try It Out
• This gives us the accumulated area under the
curve on the interval [0, b]
Applications
• Find the accumulation function for
• The surface of a machine part is the region
between the graphs of y1 = |x| and
y2 = 0.08x2 +k
• Evaluate
• Determine the value for k if the two functions
are tangent to one another
• Find the area of the surface of the machine
part
• F(0)
• F(4)
• F(6)
Revolving a Function
Volumes – The Disk Method
Lesson 7.2
• Consider a function f(x)
f(x)
on the interval [a, b]
a
• Now consider revolving
b
that segment of curve
about the x axis
• What kind of functions generated these solids
of revolution?
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Disks
Disks
f(x)
• We seek ways of using
integrals to determine the
volume of these solids
• Consider a disk which is a
slice of the solid
• To find the volume of the
whole solid we sum the
volumes of the disks
f(x)
a
b
dx
• Shown as a definite integral
• What is the radius
• What is the thickness
• What then, is its volume?
Try It Out!
Revolve About Line Not a Coordinate Axis
• Try the function y = x3 on the interval
0 < x < 2 rotated
about x-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 for
the radius
_______________
Washers
Application
• Consider the area
between two functions
rotated about the axis
f(x)
g(x)
a
• Now we have a hollow solid
• We will sum the volumes of washers
• As an integral
• Given two functions y = x2, and y = x3
• Revolve region between about x-axis
b
What will be the
limits of
integration?
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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
• Must consider curve as
x = f(y)
• Radius ____________
• Slice is dy thick
• Volume of the solid rotated
about y-axis
• How to represent the
radius?
• What is the thickness
of the disk?
Flat Washer
Assignment
• 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?
• Lesson 7.2A
• Page 465
• Exercises 1 – 29 odd
• f(y) = _____
• Radius of outer circle?
•
• Limits?
• 0 < y < 16
Cross Sections
• Consider a square at x = c
with side equal to
side s = f(c)
a c
• Now let this be a thin
slab with thickness Δx
• What is the volume of the slab?
• Now sum the volumes of all such slabs
Cross Sections
f(x)
b
f(x)
• This suggests a limit
and an integral
a
c
b
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Cross Sections
Try It Out
• Consider the region bounded
• We could do similar
summations (integrals)
for other shapes
f(x)
a
c
b
• Triangles
• Semi-circles
• Trapezoids
• 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 7.2B
• Page 4646
• Exercises 31 - 39 odd, 71, 72
Volume: The Shell Method
Lesson 7.3
Find the volume generated when
this shape is revolved about the
y axis.
If we take a
____________slice
and revolve it
about the y-axis
we get a cylinder.
We can’t solve for x, so
we can’t use a
horizontal slice
directly.
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The Shell
Shell Method
• Consider the shell as one of many of a solid of
dx
revolution
• Based on finding volume of cylindrical shells
• Add these volumes to get the total volume
f(x)
• Dimensions of the shell
f(x) – g(x)
• _________of the shell
• _________of the shell
• ________________
g(x)
• The volume of the solid made of the sum of
the shells
Try It Out!
• Consider the region bounded by x = 0,
y = 0, and
x
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
thickness = _____
_______________ = y
• What are the dimensions needed?
• radius
• height
• thickness
Rotation About Non-coordinate Axis
• Possible to rotate a region around any line
g(x)
f(x)
x=a
• Rely on the basic concept behind the shell
method
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Try It Out
Rotation About Non-coordinate Axis
• What is the radius?
r
g(x)
f(x)
• Rotate the region bounded by 4 – x2 ,
x = 0 and, y = 0 about the line x = 2
a–x
• What is the height?
x=c
x=a
f(x) – g(x)
• What are the limits?
c<x<a
• The integral:
• Determine radius, height, limits
Try It Out
Assignment
• Integral for the volume is
• Lesson 7.3
• Page 474
• Exercises 1 – 31 odd
Arc Length
Arc Length and Surfaces of
Revolution
Lesson 7.4
• We seek the distance
along the curve from
f(a) to f(b)
• That is from P0 to Pn
P0
P1
••
a
•
Pi
•
Pn
• •
b
• The distance formula for each pair of points
What is another way
of representing this?
Why?
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Arc Length
Arc Length
• We sum the individual lengths
• Find the length of the arc of the function for
1<x<2
• When we take a limit of the above, we get the
integral
Surface Area of a Cone
Assignment
• Slant area of a cone
s
h
• Lesson 7.4A
• Page 485
• Exercises 1 – 35 odd
(skip 27, 29)
r
• Slant area of
frustum
Surface Area
L
Surface Area
Δx
• Suppose we rotate the
f(x) from slide 2 around
the x-axis
• A surface is formed
• A slice gives a __________
P0
P1
••
a
• We add the cone frustum areas of all the slices
Pi
•
Pn
•
xi
• •
•
b
• From a to b
• Over entire length of the curve
Δs
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Surface Area
• Consider the surface generated by the curve
y2 = 4x for 0 < x < 8 about the x-axis
Surface Area
• Surface area =
Limitations
Assignment
• We are limited by what functions we can
integrate
• Lesson 7.4B
• Page 386
• Exercises 37 – 45 odd,
59, 66, 67
• 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
Work
• Definition
The product of
Work
Lesson 7.5
• 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
50
12 ft
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Hooke's Law
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 =
• 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
• Force required to move through i th interval, x
• W = F(xi) x
x
a
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?
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 (__________________)
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 – Guidelines
r
• Draw a picture with the
b
coordinate system
a
• Determine _______of thin
horizontal slab of liquid
• Find expression for work needed to lift this
slab to its destination
• Integrate expression from bottom of liquid to
the top
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Assignment
• Lesson 7.5
• Page 495
• Exercises 1 – 11 odd
17, 19, 25, 27, 31
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 densities
water
62.4 lbs/ft3
mercury
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 Pressure
Fluid Force on Submerged Object
• Consider a rectangular metal sheet measuring
2 x 4 feet that is submerged in 7 feet of water
• Remember
• Consider the force of fluid
against the side surface of the container
• Pressure at a point
• Density x g x depth
• Force for a horizontal slice
so P = 62.4 x 7 = 436.8
• And F = P x A
so F = 436.8 x 2 x 4 = 3494.4 lbs
• Density x g x depth x Area
• Total force
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Assignment
Fluid Pressure
• The tank has cross section
of a trapazoid
• Filled to 2.5 ft with water
• Water is 62.4 lbs/ft3
(-4,2.5)
(4,2.5)
2.5 - y
(-2,0)
(2,0)
• Lesson 7.7
• Page 513
• Exercises 1-25 odd
• Function of edge
• Length of strip
• Depth of strip
• Integral
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