The definite integral- area under a curve, Mathematics

Assignment Help:

The Definite Integral

Area under a Curve

If there exists an irregularly shaped curve, y = f(x) then there is no formula to find out the area under the curve between two points x = a and x = b on the horizontal axis. If this interval [a, b] is broken into 'n' subintervals [x1, x2], [x2, x3] ... [xn-1, xn] and rectangles are constructed in such a way that the height of each rectangle is equal to the smallest value of the function in the subinterval then the sum of the areas of the rectangles i.e.  158_area under the curve.png will approximate the actual area under the curve, where  642_area under the curve1.png , is the difference between any two consecutive values of x. The smaller the value of  642_area under the curve1.png the more rectangles can be created and the closer is the sum of the areas of the rectangles so formed, i.e.  158_area under the curve.png , to the actual area under the curve. If the number of subintervals increases, that is 'n' approaches infinity, each subinterval becomes infinitesmally small and the area under the curve can be expressed as

Area, C = 778_area under the curve2.png

Figure 1

435_area under the curve3.png

Figure 2

379_area under the curve4.png

The area under the graph of a continuous function between two points on the horizontal axis, x = a and

x = b, can be best described by the definite integral of f(x) over the interval x = a to x = b. This is mathematically expressed as

1832_area under the curve5.png 

a and b on the left hand side of the above expression are called the upper and lower limits of the integration. Unlike the indefinite integral which represents a family of functions as it includes an arbitrary constant, the definite integral is a real number which can be found out by using the  = 

fundamental theorem and is expressed as  1298_area under the curve6.png

Related Discussions:- The definite integral- area under a curve

How many packets of the first type did she purchase, The manager of a garde...

The manager of a garden store ordered two different types of marigold seeds for her display. The first type cost her $1 per packet and the second kinds cost $1.26 per packet. How m

.fractions, what is the difference between North America''s part of the tot...

what is the difference between North America''s part of the total population and Africa''s part

Group automorphism, (a) Find an example of groups G, H, K with K  H and H...

(a) Find an example of groups G, H, K with K  H and H G but K G. (b) A subgroup H of G is characteristic if σ(H) ⊆ H for every group automorphism σ of G. Show that eve

Division, Why do we start dividion operation from left to right?

Why do we start dividion operation from left to right?

Normal distribution to approximate binomial distribution, Survey 83% of com...

Survey 83% of community for a park. Randomly select 21 people if they do or do not want a park. Can you use normal distribution to approximate binomial distribution?If so find mean

Determine if r is equivalence relation or a partial ordering, Let R be the ...

Let R be the relation on the set of ordered pairs of positive integers such that ((a, b), (c, d)) ∈ R if and only if ad = bc.  Determine whether R is an equivalence relation or a p

Describe the basic concepts and terminology, Describe the Basic Concepts an...

Describe the Basic Concepts and Terminology? Somebody tells you that x = 5 and y = 3. "What does it all mean?!" you shout. Well here's a picture: This picture is what's

Prove the equality of axiom choice, (1) Prove that Zorn's lemma is equivale...

(1) Prove that Zorn's lemma is equivalent to axiom of choice. (2) Use Zorn's Lemma to prove the existence of E.

Decision-making under conditions of risk, Decision-making Under Conditions ...

Decision-making Under Conditions of Risk With decision-making under conditions of risk all possible states of nature are known and the decision maker has sufficient knowledge

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