Answer to Question #276353 in Calculus for khan

Question

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

Solution:

a) Differentiation is the process of finding the derivative of a function while Integration is the process of finding the integral of a function.

In differential calculus you find the derivative of a function, which gives you the rate of change of the function at any point, or in other words you get the gradient of the tangent to the curve at a point. Applications include finding maximum and minimum values.

In integral calculus, you do the opposite operation to differentiation, i.e. finding anti-derivative of a function. In other words, integration allows you to take the sum of many infinitesimal elements. Applications include finding area below a graph, volume of revolution of a function, etc.

Differences between Differentiation and Integration:

1)Purpose and Functions of Differentiation and Integration: Integration and differentiation can be primarily be differentiated in the way the two concepts are applied and their ultimate results. They are used to arrive at different answers, which is the fundamental difference. Differentiation is used in calculating the gradient of the curve. Nonlinear curves have different slopes at any given point, which makes it difficult to determine their gradients. The algebraic expression used to determine the change incurred from one point to another with a unit is referred to as differentiation. On the other hand, integration is an algebraic expression used in calculating the area under the curve because it is not a perfect shape after which area can easily be calculated.

2)Real Life Application for Differentiation and Integration : In real life scenarios, integration and differentiation have been found to be applied differently to each concept used in providing different results. Nevertheless, it is remarkable to highlight that both differentiation is essential calculus concepts that make life easy. One of the main application of integration is calculating the areas of curved surfaces, calculating the volume of objects, and calculating the central point among other functions.

On the other hand, differentiation concept is significantly used in calculating instantaneous velocity and used in determining whether a function is increasing or decreasing accordingly. This is a clear demonstration of how the two concepts are applied in the lives of individuals.

3) Directly Opposite : Differentiation and Integration algebraic functions are directly opposite of one another, specifically in their application. If one performs integration, he or she is said to be showing the opposite of differentiation while if one performs differentiation, he or she is performing opposite of integration. For example, integration and differentiation form a relationship that is similarly depicted when one performs the square of a number and then finds the square root of the result. Therefore, if one wants to find the opposite of an integrated number, he or she will be required to perform the differentiation of the same number. Simply, integration is the reverse process of differentiation and vice versa.

4) Speed and Function of Differentiation and Integration : The other difference between integration and differentiation is the role they play when it comes to any given function under investigation. According to mathematicians, differentiation significantly helps in determining the speed of the function by helping in the calculation of instantaneous velocity. On the other hand, integration is concerned with determining the distanced travelled by any given function. The area under the curve is estimated to be equivalent to the distance travelled by the function. Integration algebraic expression helps in calculating the area under the curve, which amounts to the distance travelled by the function.

5) Addition and Division

The other method of comparing integration to differentiation is by specifically explaining how each function realizes its results. Integration determines the outcome of a specific function by adding the aspects associated with calculation. On the other hand, differentiation determines instantaneous velocity and the speed of the function through division.

b) Some application of Calculus in CSE :

• Computer Graphics/ Image processing and here will also be need Analytical Geometry and Linear Algebra. In this sector students need to study some Differential Geometry (which has multivariate Calculus as a minimum prerequisite). But hehe will need Calculus here even for very basics things like "Fourier Transform" or "Wavelets".
• Data Mining and related subjects.
• Robotics, where a programmer will need to model physical movements of a robot, so he will need to know partial derivatives and gradients.
• Analysis of Algorithms, where an analyzer uses the notation of limit right from the start.

c) Geometrical meaning of definite integral with figure :

Consider a curve which is above the x – axis. It is a continuous function on an interval [a, b] where all the values are positive. The area between the curve and the x-axis defines the definite integral of any continuous function. 

$\int_{a}^{b} f(x) \,dx= F(b)-F(a)$

In the above formulae, $a$ and $b$ are the limits, $\frac{d}{dx}(F(x))=f(x)$ .