The mean value theorem for integrals, Mathematics

Assignment Help:

The Mean Value Theorem for Integrals

If f(x) is a continuous function on [a,b] then here is a number c in [a,b] thus,

ab f(x) dx = f(c)(b -a)

Proof

Let's begin off by defining,

F(x) = ab f(t) dt

Because f(x) is continuous we get alreday from the Fundamental Theorem of Calculus, Part I that F(x) is continuous on [a,b], differentiable on (a,b) and as F′(x) = f(x).

Here, from the Mean Value Theorem we get that here is a number c such as a < c < b and that,

 F(b)- F(a) = F′(c) (b - a)

Though we know that F′(c) = f(c) and,

 F(b) = ab f(t) dt = ab f(x) dx                           F(a) = aa f(t) dt = 0

Therefore we get,

ab f(x) dx = f(c) (b -a)

Work

The work done by the force F(x) as by assuming that F(x) is continuous, over the range a ≤ x ≤ b is,

W = ab F(x) dx

Proof

Let's begin off by dividing the range a ≤ x ≤ b in n subintervals of width ?x and from all of these intervals select the points x1*, x2*,...., xn*.

Here, if n is large and as F(x) is continuous we can suppose that F(x) won't differ by much over each interval and therefore in the ith interval we can suppose that the force is approximately constant along with a value of F(x) ≈ F(x*). The work on every interval is then approximately,

Wi ≈ F(xi*) ?x

The complete work over a ≤ x ≤ b is approximately then,

2170_mean1.png

At last, if we take the limit of that as n goes to infinity we will find the exact work done. Therefore,

1887_mean2.png

It is, though, nothing more than the definition of the definite integral and therefore the work done through the force F(x) over a ≤ x ≤ b is,

W = ab F(x) dx


Related Discussions:- The mean value theorem for integrals

Shares and dividend, a man in rested rupee 800 is buying rupee 5 shares and...

a man in rested rupee 800 is buying rupee 5 shares and then are selling at premium of rupee 1.15. He sells all the shares.find profit

Substitute 6 for r in the formula a = r^2 and solve for a, Find the area of...

Find the area of a circle along with a radius of 6 inches. The formula for the area of a circle is A = πr 2 . Use 3.14 for π. Substitute  6 for r in the formula A = πr 2 and solve

Learning and formulating maths teaching strategies, Before going further, l...

Before going further, let us repeat an aspect of learning which is useful to keep in mind while formulating teaching strategies. A child who can add or subtract in the context of s

Estimation of difference among two means-illustration, A comparison of the ...

A comparison of the wearing out quality of two types of tyres was obtained by road testing. Samples of 100 tyres were collected. The miles traveled until wear out were recorded and

Statistics, How many 4 digit numbers can be formed using the numbers: 1 – 7...

How many 4 digit numbers can be formed using the numbers: 1 – 7. Repeated numbers CAN NOT be used

Example of integration by parts - integration techniques, Example of Integr...

Example of Integration by Parts - Integration techniques Illustration1:  Evaluate the following integral. ∫ xe 6x dx Solution : Thus, on some level, the difficulty

Linear programming, #question.areas of applications of linear program mes t...

#question.areas of applications of linear program mes to solution to engineering problems.

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