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7.1 Area Between 2 Curves
Objective: To calculate the area between 2 curves.
Type 1: The top to bottom curve does not change.
f(x)
*Vertical rectangles top – bottom gives the height
g(x)
a
b
This will sum the rectangles from the left to the right with width ∆x.
Ex: Find the area bounded by y = x and y = 4x – x2.
Type 2: The top to bottom curve change within the given interval.
g(x)
c
A=
∫
a
a
b
b
f(x) − g(x) dx +
∫ g(x) − f(x) dx
c
f(x)
If the top to bottom curve change within the interval, you must start a
new integral. You will also need to calculate the point of
intersection of the 2 curves!!!
Ex: Find the area bounded by the curves y = x2 and y = 2 - x2 from [-2, 1].
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Type 3: If you need to use horizontal rectangles….
g(y)
f(y) * Now (right – left) gives height and all
d
equations must be in terms of y. You will
also integrate from the bottom to the top.
You will also need the y-value of the point of
intersection.
c
Ex: Find the area with respect to the y-axis bounded by the curves
y = 2x and y = x3 from y = -2 to y = 1.
Could you calculate this area to the x-axis?
Ex: Find the area bounded by a) y = x2 + 1 and y = 2
b) x = y2 and y = x - 2
Can you calculate the area between 2 curves?
Assign 7.1: Set up, but do not integrate →1-6, 17-18, 19-27odd, 31,
33, 37-41odd, 61, 69
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7.2a Volumes of Revolution
Objective: To calculate the volume of a solid when a region is
revolved about the x or y-axis. ☺ (Disk Method)
Disks to the x-axis: This means you are revolving your region
about the x-axis.
f(x)
•If f(x) is revolved about the ______the
solid created has solid circular cross
sections with ____________________
•Rectangle ⊥ to the x-axis ⇒ this is
the radius of the circle which is the
height of the curve, f(x).
a
b
•Integrate from a to b (left to right)
along the x-axis
Ex: Find the volume created when y = x2 is revolved about the
x-axis from [0, 2].
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Disks to the y-axis: This means you are revolving your region
about the y-axis.
•If f(x) is revolved about the ______the
solid created has solid circular cross
sections with __________________.
•Rectangle ⊥ to the y-axis ⇒ this is
the radius of the circle which is the
height of the curve, f(y).
•Integrate from c to d (bottom to top)
along the y-axis
Ex: Find the volume created when y = x2 is revolved about the
y-axis from y = 0 to y = 4.
Note: This is the same equation as the previous example,
but when spun about the y- axis creates a different volume.
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Washer Method:
Method This is when you are revolving a region that is not
bounded to the x or y-axis. Your cross section looks like a washer.
What is the area of the shaded region?
Find the volume when the region
bounded by y = x2 + 1, y = x [0, 2]
is revolved about the x-axis.
x
Use what you have learned to derive the formula for the
volume of a sphere with radius r.
Hint: Let the center be (0, 0) and rotate about the x-axis.
What shape do you need? What is the equation for this shape?
Make sure you know the difference between the disk and washer
method. You must also know the difference in the set-up depending
upon the axis you revolve about (what variables should you use?)
Assign: 7-2a: 1-10 all, 11 & 12 (a and b only) 13a
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7.2B
OTHER AXES OF REVOLUTION
Objective: To determine the volume when the region is spun about
any horizontal or vertical axis.
Spin about y = c: Remember, you must look at the outer2 – inner2
with respect to the axis of revolution.
1. Above the region
2. Between
3. Opposite side of
axis
y=c
y=c
a
b
a
b
y=c
a
b
NOTE: The axis of revolution cannot pass through the region
you plan to revolve.
If you spin about a horizontal axis, all variables must be
in terms of …
Find the volume when you spin the region enclosed by y = x2 , y
= 0 and x = 2 about each horizontal line.
544 π 224 π 256π
,
,
(Set up the integral,then use your calculator.)
calculator.)
15
15
15
a) y = 8
b) y = 4
c) y = -2
6
Spin about x = c: Remember, you must look at the outer2 – inner2
with respect to the axis of revolution.
When you spin about a vertical axis, all variables must be
in terms of …
Find the volume when you spin the region enclosed by y = x2 , y
= 0 and x = 2 about each vertical line.
(Set up the integral,then use your calculator.)
calculator.)
a) x = 4
b) x = 2
c) x = -2
40π 8π 56π
,
,
3
3
3
Now try all of question #14.
Find the volume when the region bounded by y = 6 – 2x – x2 and
y = x + 6 is revolved about
a) x-axis
b) y = 3
c) y = 7
Can you set up volumes to other axes of revolution? Do you know
what variables to use for horizontal/vertical axes of revolution?
Set up / Calculator / check answer Don’t forget π!
(Ask for even answers) 7-2b: 11 & 12( b and c only), 13-22 all
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7.2c Volumes by Slicing ☺
Objective: To find the volume of a solid with a known cross section by slicing.
Questions from 6.2a or 6.2b?
Find the volume of the solid whose base is bounded by y = 2x and
y = x2 cross sections perpendicular to the x-axis are
a) squares:
b)semi-circles:
c) equilateral triangles:
base
d) rectangles with height = 2:
e) isosceles right triangles with leg = base:
f) isosceles right triangles with hypotenuse = base:
Why is it important to know that the cross sections are ⊥ to the x-axis?
NOTE:
NOTE THESE DO NOT SPIN!
SPIN
You will not have π unless it is in the area formula.
a)
a) squares:
base =
A of cross section =
V=
16
2π
, b)
15
15
b) semi-circles:
base =
A of cross section =
V=
1. What is the base? (Sketch the base and determine the length of the base.
2. What is the area of the cross section in terms of the base?
b
3. V = ∫ A(x) dx
a
8
c)
4 3
8
, d)
15
3
c) equilateral triangles:
base =
d) rectangles with height = 2:
A of cross section =
A of cross section =
V=
base =
V=
e)
8
4
, f)
15
15
Isosceles right triangles e) leg = base:
f) hypotenuse = base:
base =
base =
A of cross section =
V=
A of cross section =
V=
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There is a short cut here.
a) squares:
b) semi-circles:
c) equilateral triangles:
d) rectangles with height = 2:
e) isosceles right triangles with leg = base:
f) isosceles right triangles with hypotenuse = base:
Cross Sections on Circular Bases
Find the volume of the solid whose base is the region bounded by
x2 + y2 = 4 if cross sections ⊥ to the x-axis are This is the base of your shape. Find the base in
terms of x if ⊥ the x-axis and y if ⊥ to the y-axis.
a) Squares
A=
V=
b) isosceles right ∆-leg base
A=
V=
10
You try these!
a)
2048 3
, b) 603.185
3
Find the volume of the solid whose base is the region bounded by
x2 + y2 = 64 if cross sections ⊥ to the x-axis are a) Equilateral triangles
b) rectangles with h = 3
Can you determine volumes by slicing? Do you know when you
need π?
7-2c: 23, 24, 27, 57-59**, 63, 71, 72, 74
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