Continuous Integration: definition, types, and solved questions

Continuous integration is a method of adding small parts of a function to find the whole. Integration and differentiation are the two major operations in the field of calculus. Integration is a way to take the sum of very small sections to find the whole section. Integration is the reverse process of differentiation. It has a lot of applications from finding the area of asymmetric shapes and regions bounded by different mathematical functions. Continuous integration is assigning numerical values to some mathematical functions, which has some minimal value for it. We can find numerous parameters such as displacement, volume, area, energy, etc. 

Read: Differentiation and Integration Formula

Key takeaways: Continuous integration, calculus, differentiation, Basics of continuous integration, types of integration, definite integral, indefinite integral

---

What is Continuous Integration?

[Click Here for Sample Questions]

By assigning numerical values to the function, it helps us to represent various parameters such as displacement, area, volume, etc. Integrals and differentials are the backbones for many calculus applications. Continuous integrals are one of them. Continuous integrals are helpful in so many ways. Earlier, areas of only defined shapes could be found using algebra but after the development of calculus and continuous integration, we can find the area of any kind of curve given to us the function of that curve. We can find areas, volumes of many asymmetric objects just by knowing the function of that curve using continuous integration.

Also read: Integration by Partial Fractions

Integrals Detailed Video Explanation

---

Basics of Continuous Integration

[Click Here for Sample Questions]

When we integrate tiny pieces of data and assign it to a function, it represents something meaningful like displacement, area, volume. Integration and differentiation are two important aspects of calculus and we also know that integration and differentiation are two opposite things operations on a function. 

 Let us understand it using an example. Consider a function f(x) which is continuous in the interval a<= x <= b and let h(x) be a function such that d h(x)/dx = f(x) for every x in (a,b).

Let the interval [a,b] be I.

Now h(x) can be defined by 

h(x) = ab f(x)dx  

In general, 

A function F(x) can be defined by 

F(x) = axf(t)dt

 Let c be in I

Let x be very close to c in the interval I,

acf(t)dt = axf(t)dt + xcf(t)dt

acf(t)dt - axf(t)dt = xcf(t)dt

F(c) - F(x) = xcf(t)dt

So in continuous integration, let f(y) = y , u(k) = a, v(k) = k

In the case here, the function is independent of k and it is supposed to be substituted with f,u,v

 Also read Integration by Parts

---

Types of Integration

[Click Here for Sample Questions]

In general, there are only two types of integrals definite and indefinite integral. The fundamental theory of integral is finding large asymmetrical areas and finding areas under the curve for different functions.

Also Read Application of Integrals

Definite integrals: We can define definite integral to be exactly the value of area under the curve for the function within the given limits. All the integrals with limits given fall under definite integrals. Two values are required to know the limits of integration. One is the lower limit and the other is the upper limit. Let a be the lower limit and b be the upper limit. It is not necessary that b should be greater than a or a should be less than b. Both a and b combined can be called the limits of integration. 

The definite integral as a limit of sum:

The definite integral abf(x)dx is the area bounded by the curve y= f(x) , the ordinates x = a and x=b and the x axis is given by

abf(x)dx = (b-a) n∞1/n[ (f(a) + f(a+h) + ….. f (a +(n-1)h) ]

abf(x)dx = h0h[ (f(a) + f(a+h) + ….. f (a +(n-1)h) ]

Where h = (b-a) / n as n -> ∞ 

Indefinite Integrals: It is an integral of a function whose limits are not provided. This is the reverse method for differentiation in calculus. Its value is not always exact, that's why it has an arbitrary constant. It signifies a sense of ambiguity.

Some properties of Indefinite Integral:

1. The process of differentiation and integration are reverse of each other,

 d/dxf(x)dx = f(x) and f’(x)dx = f(x) + c

Where c is any arbitrary constant

2. Two indefinite integrals with same derivative lead to the same family of curves and so are equivalent . So if f and g are two functions such that 

 d/dxf(x)dx = d/dxg(x)dx then f(x)dx and g(x)dx are equivalent.

3. The integral of the sum of two function equals to the sum of integrals of function i.e,

(f(x)+g(x))dx = f(x)dx + g(x)dx

4. A constant factor may be written either before or after the integral sign i.e,

af(x)dx = a f(x)dx , where a is any constant

Also read: Methods of Integration

---

Things to Remember

[Click Here for Sample Questions]

1. The fundamental theory of integral is finding large asymmetrical areas and finding areas under the curve for different functions.
2. There are only two types of integrals definite and indefinite integral.
3. All the integrals with limits given fall under definite integrals.
4. The process of differentiation and integration are reverse of each other, d/dxf(x)dx = f(x) and f’(x)dx = f(x) + c
5.  It is not necessary that upper limit should be greater than lower limit or vice versa
6. Indefinite integral is an integral of a function whose limits are not provided.

Also Read:

---