maths > integral-calculus

Integral Calculus: Understanding Application Scenarios

what you'll learn...

Overview

The application scenarios of integrals are explained in detail with examples.

»  cause-effect relation in quantities.

»  effect is the "aggregate" of cause

eg: displacement is "continuous-aggregate" of speed

cause-effect relation

One of the fundamental aspects of science is to measure and specify quantities. Some examples are

•  length of a pen is $10$$10$cm

•  mass of an object: $20$$20$ gram

•  temperature of water: ${30}^{\circ }$${30}^{\circ}$ Celsius

•  the amount of time taken: $3$$3$ seconds

•  the amount of distance traveled: $20$$20$ meter

•  the speed of a car : $20$$20$meter per second

A pen can be used to write $30$$30$ pages. With $4$$4$ pens, one can write $4×30=120$$4 \times 30 = 120$ pages. Increase in the number of pen causes increase in the number of pages, which is the effect of the cause.

In this "number of pen" is a cause and "number of pages" is an effect.

This is an example of cause and effect relation.

This is a brief on "relations and functions".

Some cause-effect relations are

•  Volume of Paint and painted area

•  Number of tickets sold and the money collected in the sale

•  speed of a car and distance covered in an hour

2 liter of paint is required to paint 3 square meter. If 14 liter paint is available, how much area can be painted?

The answer is "$14×\frac{3}{2}$$14 \times \frac{3}{2}$"

•  The "volume of paint" is the cause.

•  The "area painted" is the effect.

•  This cause-effect relation is defined by a function involving multiplication by a constant.
$\text{area}=\text{volume}×\frac{3}{2}$$\textrm{a r e a} = \textrm{v o l u m e} \times \frac{3}{2}$.

Everyday, a hotel sends a worker to buy eggs from market. The eggs are priced at $1$$1$ coin each and the worker charges $5$$5$ coins for the travel to buy eggs. How many coins are to be given to buy $120$$120$ eggs?

The answer is "$125$$125$ coins".

•  The "number of eggs" is the cause.

•  The "coins" is the effect.

•  This cause-effect relation is defined by a function involving addition of a constant.
$\text{coins}=$$\textrm{c o \in s} =$ $\text{number of eggs}$$\textrm{\nu m b e r o f e g g s}$ $×\phantom{\rule{1ex}{0ex}}\text{price per egg}$$\times \textrm{p r i c e p e r e g g}$ $+5$$+ 5$

A car is moving in a straight line at constant speed. It is at a distance $10$$10$m at $20$$20$sec and at a distance $20$$20$m at $25$$25$sec. The "effect" distance is given and the "cause" speed is to be computed. What is the speed?

The answer is "speed $=\frac{20m-10m}{25\mathrm{sec}-20\mathrm{sec}}$$= \frac{20 m - 10 m}{25 \sec - 20 \sec}$".

•  The speed is cause.

•  The distance traveled is the effect.

•  This cause-effect relation is defined by a function involving rate of change.

$\text{speed}=\frac{\text{speed2}-\text{speed1}}{\text{time2}-\text{time1}}$$\textrm{s p e e d} = \frac{\textrm{s p e e d 2} - \textrm{s p e e d 1}}{\textrm{t i m e 2} - \textrm{t i m e 1}}$

A car is moving in a straight line at constant speed. It has a velocity of $2$$2$ m/sec for first $3$$3$ seconds and $4$$4$ m/sec for the next $1$$1$ sec. What is the distance traveled in the $4$$4$ seconds?

The answer is "$=2m/\mathrm{sec}×3\mathrm{sec}$$= 2 m / \sec \times 3 \sec$ $\quad + 4 m / \sec \times 1 \sec$"

•  The speed is cause.

•  The distance traveled is the effect.

•  The cause-effect relation is defined by a function involving aggregate .

$\text{distance}=\text{speed1}×\text{time1}$$\textrm{\mathrm{di} s \tan c e} = \textrm{s p e e d 1} \times \textrm{t i m e 1}$$\quad + \textrm{s p e e d 2} \times \textrm{t i m e 2}$

summary

From the examples, it is understood that, Definition of a relation as an expression involves