The fundamental theorem of calculus is a powerful mathematical tool that connects the concepts of differentiation and integration. It consists of two parts: the first part, referred to as the first fundamental theorem of calculus, deals with the evaluation of definite integrals, while the second part, the second fundamental theorem of calculus, relates differentiation and indefinite integration.

The first fundamental theorem of calculus states that if a function f(x) is continuous on a closed interval [a, b], and F(x) is an antiderivative of f(x), then the definite integral of f(x) from a to b can be evaluated by taking the difference between the values of the antiderivative at the endpoints:

∫[a, b] f(x) dx = F(b) - F(a).

This theorem allows us to bypass the sometimes challenging process of finding the area under a curve by evaluating definite integrals using antiderivatives. Essentially, it connects the concept of accumulation (integration) with that of differentiation (finding antiderivatives).

The second fundamental theorem of calculus states that if we have a function f(x) that is continuous on an interval [a, b] and F(x) is any antiderivative of f(x), then:

∫[a, b] f(x) dx = F(x) | [a, b] = F(b) - F(a).

In other words, if we know an antiderivative of a function, we can find the value of the definite integral over an interval by evaluating the antiderivative at the endpoints.

Overall, the fundamental theorem of calculus provides a bridge between differentiation and integration, enabling us to calculate definite integrals using antiderivatives and vice versa. It has widespread applications in areas such as physics, engineering, economics, and other fields where calculations involving areas, accumulations, and rates of change are necessary.
The fundamental theorem of calculus is a fundamental concept in calculus that establishes a relationship between differentiation and integration. It consists of two parts: the first part, also known as the first fundamental theorem of calculus, and the second part, known as the second fundamental theorem of calculus.

The first part of the fundamental theorem of calculus states that if a function F(x) is continuous on the interval [a, b] and F(x) is an antiderivative of f(x) on that interval, then the definite integral of f(x) from a to b is equal to F(b) minus F(a). In other words, if we find an antiderivative of a function and evaluate it at the upper and lower limits of integration, we can find the value of the definite integral.

Mathematically, the first part can be stated as:

∫[a, b] f(x) dx = F(b) - F(a)

This theorem greatly simplifies the calculation of definite integrals, as it allows us to bypass the process of finding the limit of a Riemann sum and instead evaluate the antiderivative at the limits.

The second part of the fundamental theorem of calculus establishes that if F(x) is any antiderivative of f(x) on an interval [a, b], then the definite integral of f(x) from a to b is equal to F(x) evaluated at the upper limit of integration minus F(x) evaluated at the lower limit of integration. Mathematically, it can be stated as:

∫[a, b] f(x) dx = F(x)|[a, b] = F(b) - F(a)

This part of the theorem highlights the connection between differentiation and integration, where the derivative of an antiderivative yields the original function.

The application of the fundamental theorem of calculus is vast and essential in many areas of mathematics, physics, and engineering. It enables us to calculate areas, volumes, and lengths of curves, solve differential equations, analyze motion, estimate accumulated quantities, and much more. It forms the foundation for many other advanced concepts in calculus and is crucial for understanding and applying calculus in various real-world scenarios.