Interpretations of definite integral, Mathematics

Assignment Help:

Interpretations of Definite Integral

There are some quick interpretations of the definite integral which we can give here.

Firstly, one possible interpretation of the definite integral is to give the net area among the graph of f (x) and the x-axis on the interval [a,b].  Therefore, the net area among the graph of

f ( x ) = x2 + 1 and the x-axis on [0,2] is,

                                              ∫02  x2  + 1dx=14/3

This was the exact area which was given for the initial set of problems that we looked at in this area.

Another interpretation is sometimes called the Net Change Theorem.  This interpretation says that if  f( x ) is some quantity (hence f ′ ( x ) is the rate of change of f ( x) , then,

                                                     ∫ab  f ′ ( x ) dx = f (b ) - f ( a )

is the net change in f ( x )on the interval [a,b].  In other terms, calculate the definite integral of a rate of change & you'll obtain the net change in the quantity.  We can illustrates that the value of the definite integral, f (b ) - f ( a ) , does actually give use the net change in f ( x ) and therefore there actually isn't anything to prove with this statement. It is actually just an acknowledgment of what the definite integral of a rate of change tells us.

Therefore as a quick example, if V (t ) is the volume of water within a tank then,

29_Definite Integral3.png

is the total change in the volume as we go from time t1  to time t2 .

Similarly, if s (t ) is the function giving the position of some of the object at time t we know that the velocity of the object at any time t is : v (t ) = s′ (t ) . Thus the displacement of the object time t1  to time t2 is,

1093_Definite Integral4.png

Note as well that in this case if v (t ) is both positive & negative (that means the object moves to both the right & left) in the time frame it will NOT give the net distance traveled.  It will just give the displacement that means the difference amongst where the object started and where it ended up. To obtain the total distance traveled by an object we'd ought to compute,

732_Definite Integral5.png

It is significant to note here that the Net Change Theorem only actually makes sense if we're integrating derivative of a function.


Related Discussions:- Interpretations of definite integral

Rational Number Application, in the horizontal bar event the u.s.a scored 2...

in the horizontal bar event the u.s.a scored 28.636,gremany scroed 28.7,romnia scored 27.962,and chain scored 28.537 points.which list shows these scored in descending order

Infinite limits, Infinite Limits : In this section we will see limits who...

Infinite Limits : In this section we will see limits whose value is infinity or minus infinity.  The primary thing we have to probably do here is to define just what we mean w

How much did he have in savings at the starting, Bill spent 50% of his savi...

Bill spent 50% of his savings on school supplies, and then he spent 50% of what was left on lunch. If he had $6 left after lunch, how much did he have in savings at the starting?

Compound angles, determine the exact value of cos (11*3.145/6)

determine the exact value of cos (11*3.145/6)

Sequence and series, how can we prove that an absolute convergent series is...

how can we prove that an absolute convergent series is convergent but the converse is not true.

Describe the properties of inequalities, Describe the Properties of Inequal...

Describe the Properties of Inequalities ? Postulate In comparing two quantities, say a and b, there are exactly three possibilities. (1) a is less than b. (a b)

Discrete mathmatics, give an example of a relation R that is transitive whi...

give an example of a relation R that is transitive while inverse of R is not

Proportional Relationships, Carmen bought 3 pounds of bananas for $1.08. Ju...

Carmen bought 3 pounds of bananas for $1.08. June paid for her purchase of bananas. If they paid the same price per pound, how many pounds did June buy?

Find out the area of the region, Find out the area of the region enclosed b...

Find out the area of the region enclosed by y = x 2 & y =√x . Solution Firstly, just what do we mean by "area enclosed by". This means that the region we're interested in

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd