Difference Between Integration and Differentiation

Integration vs Differentiation

Integration and Differentiation are two fundamental concepts in calculus, which studies the change. Calculus has a wide variety of applications in many fields such as science, economy or finance, engineering and etc.

Differentiation

Differentiation is the algebraic procedure of calculating the derivatives. Derivative of a function is the slope or the gradient of the curve (graph) at any given point. Gradient of a curve at any given point is the gradient of the tangent drawn to that curve at the given point. For non linear curves, the gradient of the curve can vary at different points along the axis. Therefore, it is difficult to calculate the gradient or the slope at any point. Differentiation process is useful in calculating the gradient of the curve at any point.

Another definition for derivative is, “the change of a property with respect to a unit change of another property.”

Let f(x) be a function of an independent variable x. If a small change (∆x) is caused in the independent variable x, a corresponding change ∆f(x) is caused in the function f(x); then the ratio ∆f(x)/∆x is a measure of rate of change of f(x), with respect to x. The limit value of this ratio, as ∆x tends to zero, lim_∆x→0(f(x)/∆x) is called the first derivative of the function f(x), with respect to x; in other words, the instantaneous change of f(x) at a given point x.

Integration

Integration is the process of calculating either definite integral or indefinite integral. For a real function f(x) and a closed interval [a, b] on the real line, the definite integral, _a∫^b f(x), is defined as the area between the graph of the function, the horizontal axis and the two vertical lines at the end points of an interval. When a specific interval is not given, it is known as indefinite integral. A definite integral can be calculated using anti-derivatives.

What is the difference between Integration and Differentiation?

The different between integration and differentiation is a sort of like the difference between “squaring” and “taking the square root.” If we square a positive number and then take the square root of the result, the positive square root value will be the number that you squared. Similarly, if you apply the integration on the result, that you obtained by differentiating a continuous function f(x),it will leads back to the original function and vice versa.

For example, let F(x) be the integral of function f(x)=x, therefore, F(x)=∫f(x)dx = (x²/2) + c, where c is an arbitrary constant. When differentiating F(x) with respect to x we get, F’ (x)=dF(x)/dx= (2x/2) + 0 = x, therefore, the derivative of F(x) is equal to f(x).