Fundamental Theorem of Calculus: Part 1 and 2

The fundamental theorem of calculus is a powerful tool in mathematics that establishes a deep connection between the derivative and integral of a function. In this article, we will explore the two parts of the theorem, which are crucial to understanding the relationship between these two fundamental concepts.

Part 1: The Area Under the Curve

When considering the area under a curve, it is natural to assume that the width of the interval should be zero since there is no physical width. However, mathematically, we can calculate the area at this point, say x = a, using the formula:

F(x) = ∫a x f(t)dt

where F(x) is the antiderivative of f(x) on the interval 'a, x'. The derivative of F(x), F'(x), is then given by:

F'(x) = d/dx ∫a x f(t)dt = f(x)

This shows that the derivative of the area under the curve is equal to the function itself. This fundamental result has far-reaching implications in many areas of mathematics and science.

Part 2: The Mean Value Theorem for Integrals

In Part 1, we showed that the area under a curve can be calculated using the antiderivative of the function. However, what happens when we consider the area under the curve between two points, say x = a and x = b? Using our Mean Value Theorem for Integrals, we can rewrite this equation as:

F'(x) = lim h→0 1/h ∫x x+h f(t)dt

This formula allows us to calculate the area under any two points of a curve. For example, if we want to find the area between points a and b, we can use the formula:

Area = width × average height of the function on 'a, b'

Using this formula, we can calculate the area under any curve, which is a fundamental result in calculus.

In conclusion, the fundamental theorem of calculus establishes a deep connection between the derivative and integral of a function. Part 1 shows that the area under a curve can be calculated using the antiderivative of the function, while Part 2 uses the Mean Value Theorem for Integrals to calculate the area under any two points of a curve. These results have far-reaching implications in many areas of mathematics and science.

Fundamental theorem of calculus part 1 and 2