Integral Calculus is the branch of mathematics that studies integrals and their properties. Integration is a key concept since it is the inverse process of differentiation. The fundamental theorem of calculus connects both integral and differential calculus.
Let’s learn what is Integral Calculus, its types, formulas, methods, examples, and applications of integrals in detail.
Integral Calculus
We can calculate f if we know the f’ of a differentiable function in its domain. We used to call f’, the derivative of the function f, in Differential Calculus. The anti-derivative or primal of the function f’ is referred to as f in Integral Calculus. Anti-differentiation or integration is the process of locating anti-derivatives. It is, as the name implies, the inverse of finding differentiation.
Example:
Given: f(x) = x2 .
Derivative of f(x) i.e., f'(x) = 2x = g(x)
if g(x) = 2x, then anti-derivative or integral of g(x) = ∫ g(x) = x2
Types of Integrals
There are two types of integrals, namely,
Definite Integral
Indefinite Integral
Definite Integral
A definite integral is one that contains the upper and lower limits (i.e., start and finish values). When the value of x is constrained to lie on a real line, a definite Integral is also known as a Riemann Integral.
A definite Integral is written as:
∫ab f(x) dx
Indefinite Integral
The upper and lower bounds are not used to define indefinite integrals. The indefinite integrals reflect the derivatives of the supplied function, which returns a function of the independent variable.
F(x) denotes the integration of a function f(x) and is represented by:
∫f(x) dx = F(x) + C
R.H.S. is the integral of f(x) with regard to x.
F(x) is the anti-derivative or primitive.
f(x) is the integrand.
dx is the integrating agent.
C is the constant of integration.
and,
x is the integration variable.
Methods to Find Integrals
There are various methods for calculating indefinite integrals. The most common methods are:
- Integration by substitution method.
- Integration by parts.
- Integration by partial fractions.
Finding Integrals by Substitution Method
A few integrals are found using the substitution method. If u is a function of x, u’ = du/dx.
∫ f(u)du = ∫ f(u)u’ dx,
where u = g(x).
Finding Integrals by Integration by Parts
Integrals are obtained using the method of integration by parts when two functions are of the product form.
f(x)∫ g(x) dx – ∫ (f'(x) ∫g(x) dx) dx = ∫f(x)g(x) dx.
Finding Integrals by Integration by Partial Fractions
Integration of rational algebraic functions with positive integral powers of x in the numerator and denominator and constant coefficients is achieved by resolving them into partial fractions.
Decompose this improper rational function to a valid rational function and then integrate to determine f(x)/g(x) dx.
∫f(x)/g(x) dx = ∫ p(x)/q(x) + ∫ r(x)/s(x), where g(x) = a(x) . s(x)
Application of Integrals
There are many applications of integrals some of which are listed below:
In Maths,
- To locate the centre of mass (Centroid) of a curved-sided area.
- To calculate the area between two curves.
- To calculate the area under a curve.
- A curve’s average value.
In Physics
Integrals are used to calculate:
- The gravitational centre.
- Vehicle mass and momentum of inertia
- Satellite mass and motion.
- A tower’s mass and momentum.
- The centre of mass.
- The velocity of a satellite when it is launched into orbit.
- The trajectory taken by a satellite as it is sent into orbit.
- In calculating Thrust.
Furthermore, the real-world application of integrations is determined by the industry kinds in which this calculus is employed. Engineers use integrals to determine the geometry of building structures or the power line required to connect the two substations, among other things.