Meaning of Integration: In a broad sense, in calculus, the idea of limit is used where algebra and geometry is implemented. Limits helps us study the result of points on a graph get closer to each other until their distance is almost zero. We know that there are two major types of calculus –
- Differential Calculus
- Integral Calculus
We are talking about Integral Calculus here. According to Mathematician Bernhard Riemann,
“Integral is based on a limiting procedure which approximates the area of a curvilinear region by breaking the region into thin vertical slabs.”
Let us now try to understand what does that mean:
Take an example of a slope of line in a graph to see what differential calculus is:
In a slope of a line, we can find the slope by using the slope formula. But what if we are given to find a area of a curve? In a curve the slope of the points varies and it is then we need differential calculus to find the slope of a curve.
You must be familiar with finding out the derivative of a function using the rules of derivative. Wasn’t it interesting? Now we are going the other way round to find the original function using the rules in Integration. We are going to use the same rule backwards to find out the original function. In other words, Integration is the opposite of Differentiation.
There are mainly two types of Integrals:
- Definite Integral
- Indefinite Integral
- Definite Integral:
An integral that contains the upper and lower limits then it is a definite integral. On a real line, x is restricted to lie. Riemann Integral is the other name of the Definite Integral.
A definite Integral is represented as:
\(\int_{a}^{b} f(x)dx\)
- Indefinite Integral:
This is how we use Indefinite Integrals:
\(\int f(x)dx = F(x) + C\)
Where C is any constant. The function f(x) is called the integrand.
Lets Work out- Example- Find the integral of the function: \(\int_{0}^{3} x^{2}dx\) Solution- Given \(\int_{0}^{3} x^{2}dx\) = \(\left ( \frac{x^{3}}{3} \right )_{0}^{3}\) \(= \left ( \frac{3^{3}}{3} \right ) – \left ( \frac{0^{3}}{3} \right )\) \(= 9\)
Example- Find the integral of the function: \(\int x^{2}dx\) Solution- Given \(\int x^{2}dx \) = \(\left ( \frac{x^{3}}{3} \right ) + C \) |
Integration has many uses in math and in real life as well.
Use our Integration Calculator to solve any integral problem in a jiffy!
If you are looking NCERT Solutions for Class 12 Integrals, click on this link NCERT Solutions for Integrals.
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