2301107 calculus i -...
TRANSCRIPT
2301107 Calculus I7. Applications of the
integrations
Chapter 7:Applications of the integrations C07-2
Chapter 7:Applications of the integrations C07-3
Outline
7.1. Area between curves
7.2. Volume of the solid by rotation
7.3. Volume by cylindrical shells
Chapter 7:Applications of the integrations C07-4
Chapter 7:Applications of the integrations C07-5
7.1. Areas between positive curves
● Find the area of the region S that lies between positive functions f and g from a to b.
x = a
x = b
y = f(x)
S
0 x
y
a b
y = g(x)
Chapter 7:Applications of the integrations C07-6
Chapter 7:Applications of the integrations C07-7
Areas between curves
● Using Riemann sum on the range that f(x) > g(x).
S=limn∞
∑i=1
n
f xi∗−g x i
∗ x
f(x*)
g(x*)
y = f(x)
0 x
y
a b
y = g(x)
f(x*) - g(x*)
S
Chapter 7:Applications of the integrations C07-8
Chapter 7:Applications of the integrations C07-9
Areas between curves by x-axis
● The area S bounded by y = f(x), y = g(x), x=a, x=b where f and g are continuous and f(x) > g(x).
S=∫a
b
f x −g x dx
x = a
x = b
y = f(x)
S
0 x
y
a b
y = g(x)
From bottom to top
Chapter 7:Applications of the integrations C07-10
Chapter 7:Applications of the integrations C07-11
Student note
1. Find the area of the region bounded above by y = x2 + 1, bounded below by y = x and bounded on the sides by x = 0 and x = 1.
Chapter 7:Applications of the integrations C07-12
Chapter 7:Applications of the integrations C07-13
Student note
2. Find the area of the region enclosed by the parabolas y = x2 and y = 2x – x2.
Chapter 7:Applications of the integrations C07-14
Chapter 7:Applications of the integrations C07-15
Student note
3. Find the area of the whale-shaped region enclosed by the parabola y = 2 – x2 and the line y = –x.
Chapter 7:Applications of the integrations C07-16
Chapter 7:Applications of the integrations C07-17
Areas between general curves
● The area between the curves y = f(x) and y = g(x) from x = a to x = b is
S=∫a
b
∣ f x −g x∣dx
x = a
x = b
y = f(x)
0 x
y
a b
y = g(x)
S1
S2
S3
Chapter 7:Applications of the integrations C07-18
Chapter 7:Applications of the integrations C07-19
Areas between curves by y-axis
● Some regions are best treated by regarding x as a function of y. The area between x=f(y) and x=g(y) is S=∫
c
d
∣ f y−g y ∣dy
x = g(y)
0
y
c
d
x = f(y)
x
From left to right
Chapter 7:Applications of the integrations C07-20
Chapter 7:Applications of the integrations C07-21
Student note
4. Find the area enclosed by the line y = x - 1 and the parabola y2 = 2x + 6.
Chapter 7:Applications of the integrations C07-22
Chapter 7:Applications of the integrations C07-23
Student note
5. Sketch the region bounded by , x = 2 and find the area of the region.
y=1x
, y=1
x2
Chapter 7:Applications of the integrations C07-24
Chapter 7:Applications of the integrations C07-25
Student note
6. Find the area enclosed by the given curves 6.1. y = x, y = 9 – x2, x = -1, x = 2 6.2. y = x, y = x2
Chapter 7:Applications of the integrations C07-26
Chapter 7:Applications of the integrations C07-27
Student note
7. Find the area enclosed by the given curves 7.1. y = x, y = | x3 | 7.2. y = | x |, y = x2 – 2
Chapter 7:Applications of the integrations C07-28
Chapter 7:Applications of the integrations C07-29
Student note
8. Evaluate the integral and interpret it as the area of a region.
∫−1
1
∣x3−x∣dx
Chapter 7:Applications of the integrations C07-30
Chapter 7:Applications of the integrations C07-31
Volumes● Cylinder is the type of solid that can be generated
from the base and the direction of its height. Hence, the volume of the cylinder is
V = Ah where A is the area of the base and h is the height
of the cylinder.
A
r
l
h
w
V = Ah
h
V = r2h V = lwh
h
Chapter 7:Applications of the integrations C07-32
Chapter 7:Applications of the integrations C07-33
Solid● A volume of the non-cylindrical solid S can be
approximated using the cylinder of the cross-section of S. Each slice has the volume
V(Si) = A(x
i*)Dx
Therefore,
V =∑i=1
n
Axi∗ x
Area A(x
i*)
Chapter 7:Applications of the integrations C07-34
Chapter 7:Applications of the integrations C07-35
Definition of Volume
● Definition: Let S be a solid that lies between a and b. If the cross-sectional area of S in the plane P
x, through x and perpendicular to the x-axis, is
A(x), where A is a continuous function, then the volume of S is
V =limn ∞
∑i=1
n
Axi∗ x=∫
a
b
A xdx
Chapter 7:Applications of the integrations C07-36
Chapter 7:Applications of the integrations C07-37
7.2. Solid of revolution
● The solid can be formed by rotating the curve around a given axis called solid of revolution.
y = f(x)
Rotate about y-axis Rotate about x-axis
Chapter 7:Applications of the integrations C07-38
Chapter 7:Applications of the integrations C07-39
Student note
V =43
r3
9. Show that the volume of a sphere of radius r is
Chapter 7:Applications of the integrations C07-40
Chapter 7:Applications of the integrations C07-41
Student note
10. Find the volume of the solid obtained by rotating
about the x-axis the region under y = √x on [0, 1].
Chapter 7:Applications of the integrations C07-42
Chapter 7:Applications of the integrations C07-43
Student note
11. Find the volume of the solid obtained by rotating
the region bounded by y = x3, y = 8 and x = 0 about
the y-axis.
Chapter 7:Applications of the integrations C07-44
Chapter 7:Applications of the integrations C07-45
Student note
12. Find the volume of the solid that formed by rotating the enclosed region about x-axis of the curves y = x and y = x2.
Chapter 7:Applications of the integrations C07-46
Chapter 7:Applications of the integrations C07-47
Student note
13. Find the volume of the solid that formed by rotating the enclosed region about y = 2 of the curves y = x and y = x2.
Chapter 7:Applications of the integrations C07-48
Chapter 7:Applications of the integrations C07-49
Student note
14. Find the volume of the solid that formed by rotating the enclosed region about x = -1 of the curves y = x and y = x2.
Chapter 7:Applications of the integrations C07-50
Chapter 7:Applications of the integrations C07-51
Student note
15. Find the volume of a pyramid whose base is a square with side L and whose height is h.
Chapter 7:Applications of the integrations C07-52
Chapter 7:Applications of the integrations C07-53
7.3. Volume by Cylindrical Shells● The method of cylindrical shells computes the
volume by considered the difference of inner cylinder and outer cylinder.
V = V1 – V
2 = πr
22h – πr
12h
= π(r2 + r
1)(r
2 – r
1)h
= 2π(r2 + r
1)/2 h(r
2 – r
1)
V =2 r h x
r =r2r1
2
r1
r2
Chapter 7:Applications of the integrations C07-54
Chapter 7:Applications of the integrations C07-55
Volume of the solid● The volume of the solid obtained by rotating about
the y-axis the region under y = f(x) from a to b is
y = f(x)
yV =∫
a
b
2 x f xdx
2πx
f(x)
Δx
x
Chapter 7:Applications of the integrations C07-56
Chapter 7:Applications of the integrations C07-57
Student note
16. Find the volume of the solid obtained by rotating about the y-axis the region bounded by y = 2x2 - x3 and y = 0.
Chapter 7:Applications of the integrations C07-58
Chapter 7:Applications of the integrations C07-59
Student note
17. Find the volume of the solid obtained by rotating about the y-axis the region between y = x and y = x2.
Chapter 7:Applications of the integrations C07-60
Chapter 7:Applications of the integrations C07-61
Student note
18. Use cylindrical shells to find the volume of the solid obtained by rotating about the x-axis the region under the curve y = √x from 0 to 1.
Chapter 7:Applications of the integrations C07-62