INTEGRATION: PART 4

Application: Work and Pressure

Integration: Application

Area Volume

WorkEconomicsBiology

Application: Work & Pressure

dFW

Application: Work by a Variable Force

The work done by a variable force F(x) directed along the x – axis form x = a to x = b is

b

a

dxxFW )(

Work: Examples

1. A stalled car of 500 kg was pushed for 18 m. How much work was done? (weight (N)= mass (kg) . 9.8 m sec2)

2. How much work is done lifting a 1.5 kg book 2 m of the floor? (weight (N)= mass (kg) . 9.8 m sec2)

3. A 28 m uniform chain with mass of 20 kg is dangling from the roof of a building. How much work is needed to pull the chain up from the top of the building? (weight (N)= mass (kg) . 9.8 m sec2)

Hooke’s Law for Springs

Where F is force, x is the length and k is the spring constant

The force F needed to hold a spring under compression increases

linearly as the spring is compressed.

kxF

b

a

dxxFW )(

requires that a and b are the distance from the natural position of the spring.

Hooke’s Law for Springs: Examples

1. Find the work done on a spring when you compress it from its natural length of 1 m to a length of 0.75 m if the spring constant is k = 16 N/m.

2. What is the work done in compressing the spring a further 30 cm?

Hooke’s Law for Springs: Examples3. If the natural length of a spring is 0.2

m, and if it takes a force of 12 N to keep it extended 0.04 m, find the work done in stretching the spring from its natural length to a length of 0.3 m.

joules

dxxW

xxF

F

kxxF

5.1

300

300)(

12)04.0(

)(

1.0

0

Hooke’s Law for Springs: Examples

4. A force of 1200 N compresses a spring from its natural length of 18 cm to a length of 16 cm. How much work is done in compressing it from 16 cm to 14 cm?

4. During a load test, a car suspension system is lowered for 0.02 m from its no load condition when a force of 5N is applied on it. Assuming that the car suspension behaves like a spring that obeys Hooke’s Law:(a) What is the force required to lower the car for 0.01 m?(b) How much work is done to lower the car for 0.01 m?

Application: Economics

Income stream

Supply and demand curves

Consumer and producer surplus

Application: Income Stream

Find the present and future values of a constant income stream of RM100 per year over a period of 20 years, assuming that an interest rate of 10% compounded continuously.

66.864

)1(1000

1.0100

100

2

20

0

1.0

20

0

1.0

RM

e

e

dte

t

t

Present Value =

Application: Income Stream

06.6389

)1(1000

1.0100

100

100

22

20

0

1.02

1.020

0

2

20

0

)20(1.0

RM

ee

ee

dtee

dte

t

t

t

Future

Value =

Applications: Biology

World population growth

Predator-prey model

Logistic Growth models

Application: World Population Growth Implications of faster-than-exponential

population growth

r

r

r

r

tTrk

P

kdtdPP

kdtdPP

kPdt

dP

1

)1(

)1(

1

)]([

1

Application: World Population Growth This calculation shows that there is a finite

time T at which the population P goes becomes infinite -- or would if the growth pattern continues to follow the coalition model. The von Foerster paper calls this time Doomsday.

It's clear that Doomsday hasn't happened yet. To assess the significance of the population problem, it's important to know whether the historical data predict a Doomsday in the distant future or in the near future

http://www.math.duke.edu/education/ccp/materials/intcalc/worldpop/world4.html

Application: Area under the plasma curve (AUC) Uses in biopharmaceutics

(bioavailability), pharmacokinetics(drug administration) and toxicology (drug exposure and stay).