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Be seated before the bell rings DESK homework Warm-up (in your notes) Agenda : -Review - Hw p- 503 ;1-13 odd, 17-21 odd,24 - AP problem back ch 7: skip # 7
Be seated before the bell rings
DESK
homework Warm-up (in
your notes)
Agenda : -Review - Hw p- 503 ;1-13 odd,
17-21 odd,24 - AP problem back ch 7:
skip # 7
LT3: Find the volume of a non-rotational solid with known cross sections
LT3: Find the volume of a non-rotational solid with known cross sections
b
a
A x dx
Distance S 2
semi circles A= 2
r
4
2
0
dx8
V s
4
2
0
dx8
x
4
0
dx8
x
4
0
dx8
x
Area formula
2
2=
2
s
Find volume of solid with base bounded by
With perpendicular cross section to the x-axis squares
y x
b
a
A x dx
2 square = area S
4
2
0
dxV S 4
2
0
dxx
4
0
dxx 8
Find volume of solid with base bounded by y x
And
4
xy With perpendicular cross section to
the x-axis squares
2 square = area S
16
2
0
dxV S 216
0
dx4
xx
16 2
0
+ dx2 16
x x xx
16 3/2 2
0
+ dx2 16
x xx
162 5/2 3
02 5 48
x x x
1288.533
25
Find volume of solid with base bounded by y x
And
4
xy With perpendicular cross section to
the y-axis squares
2 square = area S
4
2
0
dyV S 4
22
0
4 dyy y
4
2 3 4
0
16 8y y y dy 4
3 54
0
162
3 5
y yy
34.133
Find volume of solid with base bounded by y x
And
4
xy With perpendicular cross section to
the x-axis rectangle with height 5
rect = area b h
16
0
dxV bh 16
0
5 dx4
xx
53.333
Find volume of solid with base bounded by y x
And
4
xy With perpendicular cross section to
the x-axis isosceles right triangles . With a leg on the xy plane
21
isos = 2
area s
16
2
0
1 dx
2V s
216
0
1 dx
2 4
xx
4.266
Find volume of solid with base bounded by y x
And
4
xy With perpendicular cross section to
the x-axis isosceles right triangles . With hypotenuse on xy plane
16
2
0
1 dx
4V s
216
0
1 dx
4 4
xx
2.666
21 isos
4area s
More Examples
LT1: Find the area between two curves.
Find the area of the region bounded by the two curves
LT1: Find the area between two curves.
Sketch the region bounded by the graphs of the equations and find the area of the region
LT1: Find the area between two curves.
Find the area of the region
You may use a calculator to evaluate the answer, but be sure to write the integral setup.
LT1: Find the area between two curves.
Given the area between ,x = 6, and y =0 Find the line x = a such that the area is divided into two equal regions
2y x
6
2
0
x dx6
3
0
723
x 2
0
36
a
x dx 3
03
a
x
3
363
a
3 108 4.762a
LT2: Find the volume of a rotational solid.
Find the volume of the solid created when the region bounded by
is rotated around the y axis.
~20.106
LT2: Find the volume of a rotational solid.
Find the volume of the solid created when the region bounded by
is rotated around the x axis.
4
22
0
2 8 25.132x dx
4
2 2
0
dx
LT2: Find the volume of a rotational solid.
Find the volume of the solid created when the region bounded by
is rotated around the y = 2
4
2
0
82
3x dx
LT2: Find the volume of a rotational solid.
Find the volume of the solid created when the region bounded by
is rotated around the x= 7
2
22 2
0
7 7 97.17993275y dy
2
2 2
0
dy
LT2: Find the volume of a rotational solid.
Find the volume of the solid created when the region bounded by
is rotated around the x = -1.
~36.861
2
2 2
0
dy
LT2: Find the volume of a rotational solid.
LT3: Find the volume of a non-rotational solid with known cross sections
LT1: area between two curves
Write but do not solve an integral to find the line x = k. that divides the area R in equal halves.
2
.9419440815
0
(1 cos )xe x dx
.5909624501 2 .2954812251
2
0
(1 cos ) .2954812251
k
xe x dx
2 2
1
0
(1 cos ) (1 cos )
k
x x
k
e x dx e x dx
OR
LT3: Find the volume of a non-rotational solid with known cross sections
LT3: Find the volume of a non-rotational solid with known cross sections
Write but do not evaluate an integral to find the volume of the solid whose base is R if all cross sections perpendicular to the x axis are isosceles right triangles, with a leg as a base
2
isos2
sArea
2
31
2
x
T
x edx
LT3: Find the volume of a non-rotational solid with known cross sections
Find the area bounded by the two equation from 0 and 1
.2387341293
3
0
1
3
.2387341293
x
x
e x dx
x e dx
.5353302711
LT3: Find the volume of a non-rotational solid with known cross sections
Write but do not evaluate an integral to find the volume of the solid whose base is R if all cross sections perpendicular to the x axis are semicircles.
21 semicircle
2Area r
23
2
2
x
S
T
x e
dx
LT3: Find the volume of a non-rotational solid with known cross sections
Find the volume of the solid created with R as the base if the cross sections perpendicular to the y axis are squares. You may use a calculator to evaluate the answer, but be sure to write the integral setup.
2x y 2 squareArea s
4
22
0
y dy
LT 2 Volume of rotational solid
About the y – axis
LT 2 Find the volume of the solid revolved about the given axis. And bounded by the following equation:
