Answer to Question #274974 in Calculus for Asif

Question

1. a) Define differentiation and integration in calculus. Also write down the differences between them.

b) Write down some application of Calculus in CSE.

c) Describe geometrical meaning of definite integral with figure.

Answer 1

(a)

Differentiation is the essence of Calculus. A derivative is defined as the instantaneous rate of change in function based on one of its variables. It is similar to finding the slope of a tangent to the function at a point.

Suppose you need to find the slope of the tangent line to a graph at point P. The slope can be approximated by drawing a line through point P and finding the slope by a line that is known as the secant line.

Integration is a method to find definite and indefinite integrals. The integration of a function f(x) is given by F(x) and it is represented by:

Integration is a method to find definite and indefinite integrals. The integration of a function f(x) is given by F(x) and it is represented by:

where

R.H.S. of the equation indicates integral of f(x) with respect to x

F(x) is called anti-derivative or primitive.

f(x) is called the integrand.

dx is called the integrating agent.

C is the constant of integration or arbitrary constant.

x is the variable of integration.

This integral is called indefinite integral, because the limits are not defined here.

Now for a function f(x) and any closed interval say [a,b], the definite integral is given by:

∫_a^b f(x) dx

Let us discuss here the general formulas used in integration and differentiation.

Difference between Differentiation and Integration:

Differentiation and integration both satisfy the property of linearity, i.e.,k₁ and k₂ are constants in the above equations.

1. Differentiation and Integration, both operations involve limits for their determination.
2. Both differentiation and integration, as discussed are inverse processes of each other.
3. The derivative of any function is unique but on the other hand, the integral of every function is not unique. Two integrals of the same function may differ by a constant.
4. Upon differentiating a polynomial function the degree of the result is 1 less than the degree of the polynomial function whereas in case of integration the result obtained has a degree which is 1 greater than the degree of the polynomial function.
5. While dealing with derivatives we can consider derivative at a point whereas, in the integrals, integral of a function over an interval is considered.
6. Geometrically, the derivative of a function describes the rate of change of a quantity with respect to another quantity while indefinite integral represents the family of curves positioned parallel to each other having parallel tangents at the intersection point of every curve of the family with the lines orthogonal to the axis representing the variable of integration.

(b)

In Computer Science, Calculus is used for machine learning, data mining, scientific computing, image processing, and creating the graphics and physics engines for video games, including the 3D visuals for simulations. Calculus is also used in a wide array of software programs that require it.

(c)

The area under the graph is the definite integral. By definition, definite integral is the sum of the product of the lengths of intervals and the height of the function that is being integrated with that interval, which includes the formula of the area of the rectangle. The figure given below illustrates it.