Definite Integral Notes for IIT JEE, Download PDF!
JEE Main Short Notes
Definite Integral is an important topic from the JEE Main exam point of view. Every year 1-2 questions are asked. Further, the concept of definite is used very often in different topics of JEE Main Syllabus. Daily practice is required for mastering this topic. The short notes on Definite Integral will help you in revising the topic before the JEE Main Exam. You can also download Definite Integral notes PDF at the end of the post.
1. Fundamental Theorem of Integral Calculus
Let F(x) be a function defined in [a,b] such that at all points in (a,b), F'(x)=f(x), then
If F(x) is continuous at x=b then
Similarly, if F(x) is continuous at x=a, then
2.Geometrical Significance of
Let y=f(x) be the equation of a curve referred to two rectangular axes. Let A denote the area bounded by the curve, the x axis, a fixed ordinate AG, (OA=a) and a variable coordinate MP. Let OM=x so that MP=y=f(x). The area A depends on the position of the ordinate MP whose abscissa is x and is, therefore, a function of x.
We take a point Q (x+Δx, y+Δy) on the curve which lies so near to P that, as a point moves along the curve from P to Q, its ordinate either constantly increases (shown in the first figure) or it constantly decreases (shown in the second figure).
We have ON=x+Δx, NQ= y+Δy, MN=Δx
The increment ΔA in A, consequent to the change Δx in x, is the area of the region MNQPM. The area ΔA of the figure MNQPM lies between the areas (y+Δy)Δx and yΔx of the two rectangles QM, PN.
From the first figure, we have (y+Δy)Δx>ΔA>yΔx
Let point Q tend to point P so that Δx tends to zero.
Also, we have yΔx>ΔA>(y+Δy)Δx
or, y> (ΔA /Δx)>(y+Δy)
Let point Q tend to point P so that Δx tends to zero. Thus
Let BH be the ordinate x=b. Then we have,
=The value of A (when x=b) - The value of A (when x=a)
= Area of the region GABHGA - 0
= Area of the region GABHGA
which is the area bounded by the curve y=f(x), the x-axis, x=a, and x=b.
3. Definite Integrals as the limit of Sum
Let the interval [a,b] be divided into n equal parts and let the length of each part be h so that nh=b-a. Let the function f(x) increases monotonically from a to b.
So, the entire area is divided into several trapeziums. If the length of each interval is very small, then each of these trapeziums can be approximated to be a rectangle. Again, the area of all these rectangles should be equal to the area enclosed by the curve with the positive x-axis. However, integration is for a continuous interval. To make the two areas equal, the number of divisions or intervals must be infinite or the length of each interval, h, must tend to 0.
where nh=b-a and x=r/n
The same result holds when the function f(x) monotonically decreases from a to b
Steps to express the limit of the sum as definite integral.
There are basically two steps involved in this transformation.
a. There is certain replacement required which is as follows:
|Existing||To be replaced|
b. Now put the least and the greatest values of r as lower and upper limit respectively and evaluate it further.
4. Summation of series
5. Properties of definite integrals
6. Integration using Reduction Formula
Any formula which expresses an integral in terms of another which is simpler is a reduction formula for the first integral. The successive application of the reduction formula enables us to express the integral of the general member of the class of functions in terms of that of the simplest member of the class.
The reduction formula is generally obtained by applying the rule of integration by parts.
7. Newton- Leibnitz Formula: Differentiation under the integral sign
8. Inequalities in Definite Integration
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