The Fundamental Theorem of Calculus
As we have mentioned earlier, Integral calculus helps to find the area of space under the curve formed by a function with an upper and lower limit. The calculation of said area hinges on the fundamental theorem of calculus. There are two important points in the fundamental theorem which are quite fascinating.
First, for any continuous function f [a,b] there is an antiderivative, here, we are representing it as F(x). See the equation below for more clarity. Secondly, it is this theorem that forms a connection between derivative or let say antiderivative and calculus.
[frac{dF}{dx}] = [frac{d}{dx} int_{a}^{b} f(t)dt] = [F (x)].
Here, b and a denote the boundary under the curve for which the area is to be calculated.
Categories of Integration Calculus
Here is what you should know about integration calculus. When we know the ‘f’ of a function which is differentiable in its domain, then we can then calculate the value of ‘f’. In differential calculus, the derivative of the function f is called f’. Alternately in integral calculus, we call f as the anti-derivative or primitive of the function f’.
The method of finding anti-derivatives is known as anti-differentiation or integration. Integration is the inverse of differentiation.
Integration can be Divided into Two Major Categories Which are:
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Definite Integral
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Indefinite Integral
Definite Integral
An integral that contains both the upper and lower limits that are both the start and end values, then it is recognised as a definite integral. On a real line, x is restricted to a lie. Definite Integral when restricted to lie on the real line, is also called a Riemann Integral.
A definite Integral is represented as:
[int ba f(x)dx]
Indefinite Integral
Indefinite integrals are integrals that are not defined with the upper and lower limits. Indefinite integrals represent the entire family of a certain given function whose derivatives are f. It returns a function of the independent variable.
The integration of a function f(x) is given by F(x) and it is represented by:
[int f(x) dx = F(x) + C]
where,
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R.H.S. of the equation means integral off(x) concerning x
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F(x) is called anti-derivative or primitive
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f(x) is called the integrand
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dx is called the integrating agent
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C is called constant of integration
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x is the variable of integration
It might appear unusual that there is an infinite number of anti-derivatives for a function f. An integral calculus example of the indefinite integral can make things clearer in this context.
Let us take f’ (x) = 3x². With the help of the hit and trial method, we can derive that its anti-derivative is F(x) = x³. This is due to the reason that if you differentiate F with respect to x, you will get 3x². We have only one function as the anti-derivative of f. Let us now differentiate G(x)= x³+9 with respect to x.
Again, we would get the same derivative, that is. f. This provides us with an essential insight. Since the differentiation of all constants is zero, we can write any constant with x³ and the derivative would still be equal to f.
Thus, there are infinite constants that can be substituted for c in the equation F(x) = x³+ C, and there are infinite functions whose derivative is equal to f. So, there are infinite functions whose derivatives are equal to 3 x². C is called an arbitrary constant or constant of integration.
The Uses of Integral Calculus
The two main uses or applications of integral calculus are the following:
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The calculation of f from f’. If a function f is differentiable in the interval of consideration, then f’ is defined in that interval.
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The calculation of the area under a curve.
You should refer to integral calculus PDF to get an in-depth understanding of the calculation.
Integration Calculus Examples
Below have an example of integral calculus that can come in handy for the study and understanding of the branch of calculus in a comprehensive manner:
Example – 1
[int (4x^{2} – 2x)dx= int 4x^{2}dx – int 2x dx]
=[4int xdx -2int xdx]
= [4frac{x^{3}}{3} – 2frac{x^{2}}{2} + C]
= [4frac{x^{3}}{3} – x^{2} + C]
= [x^{2}(frac{4x}{3} – 1) + C]
Example -2
[int sqrt{(4x + 4)^{frac{1}{2}}} dx]
= [frac{(4x + 4)^{2}}{4}times frac{3}{2} + C]
= [frac{(4x + 4)^{frac{3}{2}}}{6} + C]
= [sqrt{frac{(4x+ 4)^{3}}{6}} + C]
The Application of Integration Calculus
The areas where Integral Calculus is applied are as follows:
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The area between two curves
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Centre of mass
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Kinetic energy
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Surface area
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Work
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Distance, velocity, and acceleration
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The average value of a function
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Volume
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Probability