Thuyết trình toán bằng tiếng anh Ứng dụng của tích phân trong thực tế

Thai Nguyen University
Thai Nguyen University of Technology
Faculty of International Training

(MAT003)
CALCULUS II
Group 2
Supervisor : Nguyen Minh Trang
Thai Nguyen May 24th 2016


Students:

1
2
3

Lê Phước Khánh
Trần Ngọc Nam
Trịnh Ngọc Tân


CONTENT:
1, Definition of Integration
2, Application
3, Give an Example


DEFINITION OF INTEGRATION
A function F is called an antiderivative (also an
indefinite integral) of a function f in the interval I

if

F '( x) = f ( x)

for every value x in the interval I.
The process of finding the antiderivative of a
given function is called antidifferentiation or
integration.


APPLICATIONS OF INTEGRATION

Applications to
Physics and Engineering
In this section, we will talk about:
The applications of integral calculus to
force due to water pressure.


HYDROSTATIC FORCE AND PRESSURE
Suppose that a thin plate with area A m2 is
submerged in a fluid of density ρ kg/m3 at a
depth d meters below the surface of the fluid.


HYDROSTATIC FORCE AND PRESSURE

The fluid directly above the plate has volume
V = Ad
So, its mass is:

m = ρV = ρAd


HYDROSTATIC FORCE

Thus, the force exerted by the fluid on
the plate is
F = mg = ρgAd
where g is the acceleration due to gravity.


HYDROSTATIC FORCE
• The force F exerted by a fluid of a constant weightdensity w against a submerged vertical plane region
from x = a to x = b is
b

F = w∫ h( x ) L( x)dx
a

• Where w= ρ g, h(x) is the depth of the fluid and L(x) is
the horizontal length of the region at x.
Trinh Ngoc Tan


HYDROSTATIC FORCE AND PRESSURE
This helps us determine the hydrostatic
force against a vertical plate or wall or dam
in a fluid.

EXAMPLE


HYDROSTATIC F AND P

A dam has the shape of the trapezoid shown
below.
– The height is 20 m.
– The width is 50 m at the top and 30 m at the bottom.

Example


HYDROSTATIC F AND P

Find the force on the dam due to hydrostatic
pressure if the water level
is 4 m from the top of the dam.

Example 1


HYDROSTATIC F AND P

We choose a vertical x-axis with origin
at the surface of the water.

Example

HYDROSTATIC F AND P

The depth of the water is 16 m.
– So, we divide the interval [0, 16] into small intervals
of equal length with endpoints xi.
– We choose
xi* ∈ [xi–1, xi].

Example


HYDROSTATIC F AND P
The i th horizontal strip of the dam is
approximated by a rectangle with height Δx and
width wi

Example


HYDROSTATIC F AND P

From similar triangles,

a
10
=
*
16 − xi
20

Example

16 − xi
xi
a=
= 8−
2
2
*

or

*


HYDROSTATIC F AND
P
Hence,

wi = 2(15 + a )
= 2(15 + 8 − xi )
1
2

= 46 − xi

Example

*

*


HYDROSTATIC F AND P

If Ai is the area of the strip, then

Ai ≈ wi ∆x = (46 − xi ) ∆x
*

If Δx is small, then the pressure Pi on the i th strip
is almost constant, and we can use Equation 1 to
*
write: Pi ≈ 1000 gxi

Example


HYDROSTATIC F AND P

The hydrostatic force Fi acting on the i th strip is
the product of the pressure and
the area:

Fi ≈ Pi Ai
≈ 1000 gxi (46 − xi ) ∆x
*

Example

*


HYDROSTATIC F AND P

Adding these forces and taking the limit as
n → ∞, the total hydrostatic force on the dam is:
n

F = lim ∑1000 gxi (46 − xi ) ∆x
n →∞

*

*

i =1

16

= ∫ 1000 gx(46 − x) dx
0

16

= 1000(9.8) ∫ (46 x − x ) dx
2

0

Example

16


x 
2
= 9800  23 x −  ≈ 4.43 ×107 N
3 0

3


Thank You For Listening!!!