Hey there! We receieved your request
Stay Tuned as we are going to contact you within 1 Hour
One of our academic counsellors will contact you within 1 working day.
Click to Chat
1800-5470-145
+91 7353221155
Use Coupon: CART20 and get 20% off on all online Study Material
Complete Your Registration (Step 2 of 2 )
Sit and relax as our customer representative will contact you within 1 business day
OTP to be sent to Change
Introduction to Definite Integral as LImit of a Sum
Methods to express the infinite series as Definite Integral
Differentitation under the Integral Sign
Fundamental theorem of Calculus ( Newton-Leibnitz Formula)
Related Resources
We have already discussed the concept of definite integral in the previous sections. Definite integral is closely related to concepts like antiderivative and indefinite integrals. In this section, we shall discuss this relationship in detail.
Definite integral consists of a function f(x) which is continuous in a closed interval [a, b] and the meaning of definite integral is assumed to be in context of area covered by the function f from (say) ‘a’ to ‘b’.
An alternative way of describing is that the definite integral is a limiting case of the summation of an infinite series, provided f(x) is continuous on [a, b] i.e.,
The converse is also true i.e., if we have an infinite series of the above form, it can be expressed as a definite integral.
Express the given series in the form ∑ 1/n f (r/n)
Then the limit is its sum when n→∞, i.e. lim n→∞ h ∑1/n f(r/n)
Replace r/n by x and 1/n by dx and lim n→∞ ∑ by the sign of ∫.
The lower and the upper limit of integration are the limiting values of r/n for the first and the last term of r respectively.
Some particular cases of the above are
lim n→∞ ∑nr =1 1/n f(r/n) or lim n→∞ ∑n–1r = 0 1/n f(r/n) = ∫10 f(x)dx
lim n→∞ ∑pnr =1 1/n f(r/n) = ∫βα f(x)dx
where α = lim n→∞ r/n = 0 (as r = 1) and β = lim n→∞ r/n = p (as r = pn)
Show that (A) lim n→∞ {1/(n+1) + 1/(n+2) + 1/(n+3) + ... + ... 1/(n+n)} = ln 2.
(B) lim n→∞ 1p + 2p + 3p + ... + np/(np +1) = 1/(p +1) (p > 0)
(A) Let I = lim n→∞ (1/n+1 + 1/n+2 + 1/n+3 + ... + 1/n+n)
= lim n→∞ {1/(n+1) + 1/(n+2) + 1/(n+3) + ... + 1/(n+n)}
Now α = lim n→∞ 1/n = 0 (as r = 1)
and β = lim n→∞ r/n = 1 (as r = n)
⇒ l = ∫10. 1/1+x dx = [In (1+x)] 10
⇒ I = ln 2.
(B) 1p + 2p + 3p + ... + np/(np +1) = ∑nr=1 1p/n.np = ∑nr=1 1+n(r/n)p
Take f(x) = xp; Let h = 1/n so that as n → ∞; h → 0
∴ limn→∞ ∑nr =1 1/n f(0 + r/n)
= ∫10 f(x)dx = ∫10 xpdx
= 1/p+1
If g is continuous on [a, b] and f1 (x) and f2 (x) are differentiable functions whose values lie in [a, b], then
d/dx ∫ f2(x) f1(x) g(t)dt = g (f2(x)) f2'(x) – g (f1(x)) f1'(x)
f ‘(x) = – sin x – (x f (x) + ∫x0 f(t) dt) + x f(x)
= –sin x – ∫x0 f(t)dt
f “(x) = – cos x – f(x)
Hence, this gives f “(x) + f(x) = cos x.
____________________________________________________________________________
If a function f(x) is defined ∀x ∈ R such that
Diffrentiate w.r.t. x
g’(x) = – F(x)/x
F(x) = -x g’(x)
NOw, we know that g(a) = 0 and hence we get
______________________________________________________________________________
Determine a positive integer n < 5, such that
Integrating by parts,
…... (1)
= – (–1) – [ex]10 = 1 – (e–1) = 2 – e
From (i), I2 = ( -1)3 - 2I1 = -1 - 2(2 - e) = -5 + 2e
and I3 = (-1)4 - 3I2 = 1 - 3(-5 + 2e) = 16 - 6e Which is given .
∴ n = 3.
This theorem state that If f(x) is a continuous function on [a, b] and F(x) is any anti derivative of f(x) on [a, b] i.e. F'(x) = f (x) ∀ x ∈ (a, b), then
The function F(x) is the integral of f(x) and a and b are the lower and the upper limits of integration.
directly as well as by substitution x = 1/t. Evaluate why the answers don't tally.
= [1/2 tan–1 (x/2)]2–2 = 1/2 [tan–1(1) – tan–1 (–1)]
= 1/2 [π/4 – (–π/4)] = π/4
⇒ l = π/4
On the other hand; if x = 1/t then,
Solving this and simplifying, we get
I = – [1/2 tan–1 (2t)]1/2–1/2
= –1/2 tan–11 – (–1/2 tan–1 (–1)) = –π/8 – π/8 = –π/4
∴ I = π/4 when x = 1/t
In above two results l = -π/4 is wrong. Since the integrand ¼ + x2 > 0 and therefore the definite integral of this function cannot be negative.
Since x = 1/t is discontinuous at t = 0, the substitution is not valid (∴ I = π/4).
It is important the substitution must be continuous in the interval of integration.
________________________________________________________________________________________
then show that α = β.
Put x = 1/t ⇒ dx = –1/t2 then
=
Q1. The Leibnitz’s rule can be applied only if
(a) the function g is continuous in [a, b] and the functions f1 and f2 are differentiable functions whose values may lie within or outside [a, b].
(b) the function g is continuous in [a, b] and the functions f1 and f2 are differentiable functions whose values lie outside [a, b].
(c) the function g is continuous in [a, b] and the functions f1 and f2 are differentiable functions whose values lie in [a, b].
(d) the function g is discontinuous in [a, b] and the functions f1 and f2 are differentiable functions whose values may lie withing or outside [a, b].
Q2. Fundamental theorem of calculus states that = F(b) – F(a), where
(a) f(x) is a continuous function on [a,b] and F(x) is the derivative of f(x) on [a, b].
(b) f(x) is a continuous function on [a,b] and F(x) is the anti derivative of f(x) on [a, b].
(c) f(x) is a discontinuous function on [a,b] and F(x) is the derivative of f(x) on [a, b].
(d) F(x) is a continuous function on [a,b] and f(x) is the derivative of f(x) on [a, b].
Q3. In order to express infinite series as definite integral,
(a) sign of summation is replaced by integration.
(a) sign of integration is replaced by summation.
(a) infinite series can’t be expressed as definite integral.
(a) None of the above.
Q4. The definite integral is a limiting case of
(a) infinite series
(b) finite series
(c) infinite sequence
(d) indefinite integral.
Q5.
(a)
(b)
(c)
(d)
You may wish to refer indefinite integral.
Click here to refer the most Useful Books of Mathematics.
For getting an idea of the type of questions asked, refer the previous year papers.
To read more, Buy study materials of Definite integral comprising study notes, revision notes, video lectures, previous year solved questions etc. Also browse for more study materials on Mathematics here.
Get your questions answered by the expert for free
You will get reply from our expert in sometime.
We will notify you when Our expert answers your question. To View your Question
Definite Integration Geometrical Interpretation of...
21: Area bounded by y = g(x), x-axis and the lines...
Properties of Definite Integration Definite...
Area as Definite Integral Table of Content...
Solved Examples of Definite Integral Solved...
Solved Examples of Definite Integral Part I 11:...
Working Rule for finding the Area (i) If curve...
Objective Problems of Definite Problems Objective...