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

Determine the angle of depression to a ship, From the top of a 200 m lighth...

From the top of a 200 m lighthouse, the angle of depression to a ship in the ocean is 23 . How far is the ship form the base of the lighthouse?

Find a quadratic polynomial having a and ß, If α,β are the zeros of a Quadr...

If α,β are the zeros of a Quadratic polynomial such that α + β = 24, α - β = 8. Find a Quadratic polynomial having α and β as its zeros.

Polynomials, sum of zero of polynomial x2-2x+1is equal to sum of zero of po...

sum of zero of polynomial x2-2x+1is equal to sum of zero of polynomial x3-2x+x then find the product of all the three zero of the second polynomial

Algebraic expressions word problems, Juan is g years old and Eva is 2 years...

Juan is g years old and Eva is 2 years younger than Juan. a.Find the sum of their ages in terms of g. b.Find the sum of their ages in g years'' time,in terms of g.

Matrices, how solve the inverse matrices using the matlab?

how solve the inverse matrices using the matlab?

Surds, I wanted to know what are surds.please explain with an example.

I wanted to know what are surds.please explain with an example.

How many points did he score during his senior year, Michael scored 260 poi...

Michael scored 260 points during his junior year on the school basketball team. He scored 20% more points during his senior year. How many points did he score during his senior yea

Indices, 4n to the power 3/2 = 8 to the power minus 1/3. find the value of ...

4n to the power 3/2 = 8 to the power minus 1/3. find the value of n.

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