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

Compound interest, you have RM5O,OOO to invest,and two fund that you''d li...

you have RM5O,OOO to invest,and two fund that you''d like to invest in.The You-Risk-It Fund yields 14% interest.The Extra-Dull Fund yields 6% interest.Besause of college financial-

Union of sets, Union of Sets Venn diagram presenting the union of sets...

Union of Sets Venn diagram presenting the union of sets A and B or A?B = Shaded area is demonstrated below: A ?B = Shaded area

Solution process of linear differential equations, For a first order linear...

For a first order linear differential equation the solution process is as given below: 1. Place the differential equation in the correct initial form, (1). 2. Determine the i

Fractions, what is 1/3 + 2/9 equal

what is 1/3 + 2/9 equal

Pairs of straight lines, The equation ax2 + 2hxy + by2 =0 represents a pair...

The equation ax2 + 2hxy + by2 =0 represents a pair of straight lines passing through the origin and its angle is tan q = ±2root under h2-ab/(a+b) and even the eqn ax2+2hxy+by2+2gx+

Objectives of ones tens and more, Objectives After studying this unit, ...

Objectives After studying this unit, you should be able to 1.  evolve and use alternative activities to clarify the learner's conceptual 2.  understanding of ones/tens/hu

Differentiate inside function in chain rule, Differentiate following. f ...

Differentiate following. f ( x ) = sin (3x 2   + x ) Solution It looks as the outside function is the sine & the inside function is 3x 2 +x. The derivative is then.

Rules of logarithms, Rule 1 The logarithm of 1 to any base is 0. Pro...

Rule 1 The logarithm of 1 to any base is 0. Proof We know that any number raised to zero equals 1. That is, a 0 = 1, where "a" takes any value. Therefore, the loga

Function notation, Now we need to move onto something called function notat...

Now we need to move onto something called function notation.  Function notation will be utilized heavily throughout most of remaining section and so it is important to understand i

Example of partial fraction decomposition, Example of Partial Fraction Deco...

Example of Partial Fraction Decomposition Evaluate the following integral. ∫ (3x+11 / x 2 -x-6) (dx) Solution: The 1 st step is to factor the denominator so far as

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