WHAT IS INTEGRALS?

Calculus is the study of how things change.

In differentiation we are finding a derivative i.e. slope of tangent to the curve i.e how the curve changes at different points for a given function

In integration we are finding a original function whose derivative or slope of the tangent or change at different points is known

Slopes of curves is also can be considered as rate of change at different cases. This is called instantaneous change. Derivative of a curves tell us the  instantaneous rate of change of a curve.

A curvy function changes at different rates throughout its domain—sometimes it’s increasing quickly and the tangent line is steep (causing a high-valued derivative). At other places the curve may be increasing shallowly or even decreasing, causing the derivative to be small or negative, respectively.

Look at the graph of f(x) in Figure 

If instantaneous change is given to find the original state we need to integrate.

To find the change at different points we need to differentiate.

Calculus analyses things that change, and physics is much concerned with changes. For physics, you'll need at least some of the simplest and most important concepts from calculus.

A typical course in calculus covers the following topics:
1. How to find the instantaneous change (called the "derivative") of various functions. (The process of doing so is called "differentiation".)
2. How to use derivatives to solve various kinds of problems.
3. How to go back from the derivative of a function to the function itself. (This process is called "integration".)
4. Study of detailed methods for integrating functions of certain kinds.
5. How to use integration to solve various geometric problems, such as computations of areas and volumes of certain regions.

We know how to find the derivative of a function by different methods, Now we are going back in integration.

If we know that

 dy/dx=2x

and we need to know the function this derivative came from, then we "undo" the differentiation process. (Think: "What would I have to differentiate to get this result?")

y=x²  is ONE antiderivative of   dy/dx =2x

There are infinitely many other antiderivatives which would also work, for example:

y=x²+4

y=x²+π

y=x²+27.3

In general, we say y=x²+K is the indefinite integral of 2x. The number K is called the constant of integration.

Note: Most math text books use C for the constant of integration, but for questions involving electrical engineering, we prefer to write "+K", since C is normally used for capacitance and it can get confusing.

Notation for the Indefinite Integral

We write: ∫2x  dx=x²+K and say in words:

"The integral of 2x with respect to x equals x² + K."

The Integral Sign

The ∫ sign is an elongated "S", standing for "sum". (In old German, and English, "s" was often written using this elongated shape.) Later we will see that the integral is the sum of the areas of infinitesimally thin rectangles.
∑ is the symbol for "sum". It can be used for finite or infinite sums.
∫ is the symbol for the sum of an infinite number of infinitely small areas (or other variables).
This "long s" notation was introduced by Leibniz when he developed the concepts of integration.
GEOMETRICAL INTERPRETATION OF INDEFINITE INTEGRAL:
Let f (x) = 2x. Then ∫ f (x) dx = x² + C. For different values of C, we get different integrals. But these integrals are very similar geometrically.
Thus, y = x² + C, where C is arbitrary constant, represents a family of integrals. By assigning different values to C, we get different members of the family. These together constitute the indefinite integral. In this case, each integral represents a parabola with its axis along y-axis.

Clearly, for C = 0, we obtain y = x², a parabola with its vertex on the origin.
The curve y = x² + 1 for C = 1 is obtained by shifting the parabola y = x² one unit along y-axis in positive direction.
For C = – 1, y = x² – 1 is obtained by shifting the parabola y = x² one unit along y-axis in the negative direction.
Thus, for each positive value of C, each parabola of the family has its vertex on the positive side of the y-axis and for negative values of C, each has its vertex along the negative side of the y-axis. Some of these have been shown in the Fig
 


Let us consider the intersection of all these parabolas by a line x = a. In the Fig 7.1, we have taken a > 0. The same is true when a < 0. If the line x = a intersects the parabolas y = x²,
y = x² + 1, y = x² + 2, y = x² – 1, y = x² – 2 at P₀, P₁, P₂, P_-1, P_-2 etc., respectively, then dy / dx at these points equals 2a.
This indicates that the tangents to the curves at these points are parallel. Thus, ∫2x² dx = x + C = F_C (x) (say), implies that the tangents to all the curves y = F_C (x), C ∈ R, at the points of intersection of the curves by the line x = a, (a ∈ R), are parallel.

Further, the following equation (statement) ∫ f (x) dx = F (x) + C = y (say) , represents a family of curves. The different values of C will correspond to different members of this family and these members can be obtained by shifting any one of the curves parallel to itself. This is the geometrical interpretation of indefinite integral.
∫ f (x) dx = F (x) + C = Family of all curves which are geometrically similar and the tangents drawn to all curves by the line x=a (at any particular point) are parallel.

NOW THE QUESTIONS?

1.      Is All differentiable functions are integrable.

Ans: True because all differentiable functions are continuous and by FTC all continuous functions are integrable.

2.      Is All integrable functions are continuous.?

Ans:  This doesn't follow from the FTC, Now for this it is difficult to answer at this level. We have to learn advance mathematics

Can you integrate piecewise functions? If so then I can think of an easy counter-example. We've never talked about doing so in class.

3.      Is All integrable functions are differentiable.

Ans: Even though 1 is true this doesn't follow from it. Same difficulty as 2.

{Differentiable functions} ⊂ {Continuous functions} ⊂ {Integrable functions}

In order for some function f(x) to be continuous at x = c, then the following two conditions must be true:

i)                    f(c) is defined and the limit of f(x) as x approaches c is equal to f(c).

In order for some function f(x) to be differentiable at x = c, then it must be continuous at x = c and it must not be a corner point (i.e., it's right-side and left-side derivatives must be equal).

Continuity implies integrability; if some function f(x) is continuous on some interval [a,b], then the definite integral from a to b exists.

While all continuous functions are integrable, not all integrable functions are continuous. To understand this idea we need to study advanced mathematics i.e Reimann integral.

Hope this will satisfy your needs    : Happy learning   : by KHV, SAGAR