# Conceptual question of integration and derivation

In summary, the conversation discusses questions about how integration and derivatives work, specifically in regards to finding the area under a curve and the rate of change of a function. The fundamental theorem of calculus is mentioned as a way to understand these concepts, and it is suggested to consult a calculus book for more detailed explanations.
Okay I have several questions that have been bothering me for some time now, so bear with me please.

• How does integration work? Integrating the function f(x) between the bounds a,b finds the area under f(x) between a and b. But how does this work?If my function is f(x) = 2x and my bounds are [1,10], how come F(10) - F(1) equals the area under 2x from 1,10? (Where F(x) = x^2).
• Furthermore, why does a derivative work? The derivative of the function f(x) yields a relationship between the rate of change in f(x) and a given input. But how does this actually work?

Or put in a different way, if my function is 1x^2, why can I multiply the coefficient by the exponent to yield a function (2x) that tells me the rate at which the function will change?
• What does it mean that time is being removed (divided in a derivative) or added (multiplied in an integral)?

Okay I have several questions that ... how does this work?

Okay, God can tell you how it work. Human just try to compute and believe in conformation.

The best way to think of why it works is as follows:
the average rate of change on f(x) between x and x+h where h is some constant will be (f(x+h)-f(x))/h If we take the limit as h->0 for this expression then this calculates the instantaneous rate of change, or the derivative at x, since we are finding the rate of change between x and x+0, or x. The best way to visualise this is as a straight line between two points, A=(x,f(x)) and B=(x+h,f(x+h)), and then imagine B coming along the line closer and closer to A, the line will approach the tangent at point A, which is its rate of change at that moment.

For integration the same sort of idea can be used, this time imagine large rectangles approximating the area under a curve. As the width of these rectangles is lowered and a larger number of rectangles are used, the approximate area under the curve will become more accurate to the real area. If we take the limit as width->0 of the sum of the area of all these rectangles then you will get the area under the curve.

brahman said:
Okay, God can tell you how it work. Human just try to compute and believe in conformation.
That's an insult to mathematicians. Or are you saying mathematicians are not human?

On topic: these are questions that span a large portion of calculus, why don't you just take a look at a (good) calculus book (with proofs) and then ask more specific questions?

brahman said:
Okay, God can tell you how it work. Human just try to compute and believe in conformation.
Well, there are people, called mathematicians, who assit God in this work.

## What is the difference between integration and derivation?

Integration and derivation are two fundamental concepts in calculus. Integration is the process of finding the area under a curve, while derivation is the process of finding the slope of a curve at a given point. In simple terms, integration involves adding up infinitely small pieces of a curve, while derivation involves finding the rate of change of a curve.

## What are some real-life applications of integration and derivation?

Integration and derivation have many real-life applications, such as in physics, engineering, economics, and statistics. For example, integration can be used to calculate the volume of irregularly shaped objects, while derivation can be used to determine the optimum production level for a company.

## How do integration and derivation relate to each other?

Integration and derivation are inverse operations of each other. This means that if you integrate a function, you can find the original function by deriving it. Similarly, if you derive a function, you can find the original function by integrating it.

## What are some common techniques used in integration and derivation?

There are various techniques used in integration and derivation, such as the power rule, product rule, quotient rule, chain rule, and integration by parts. These techniques allow us to solve more complex integration and derivation problems by breaking them down into simpler components.

## How can I improve my understanding of integration and derivation?

To improve your understanding of integration and derivation, it is important to practice solving problems and familiarize yourself with the different techniques. It can also be helpful to visualize the concepts using graphs and diagrams, and to seek help from a teacher or tutor if needed.

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