Final answer:
The Fundamental Theorem of Calculus connects differentiation and integration, with the first part indicating that the derivative of an integrated function gets you the original function, and the second part showing that the definite integral of a function can be calculated by subtracting the antiderivative values at the bounds.
Step-by-step explanation:
Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus bridges the concepts of differentiation and integration, two main operations in calculus. It is divided into two main parts. The first part states that if you have a continuous function f(x) over an interval and you take its indefinite integral to obtain a function F(x), then the derivative of this new function F(x) at any point is equal to the original function f(x) evaluated at that point. This means that differentiation and integration are inverse processes.
Let's look at an example. Suppose we have the function f(x) = 2x. If we integrate f(x) from 0 to a to get F(a), and then we differentiate F(a), we will get back our original function, f(a) = 2a.
For the second part, it effectively states that if you want to calculate the definite integral of a function f(x) from a to b, you can take an antiderivative F(x) of f(x), and then subtract the value of F(a) from F(b). The result gives you the net area under the curve of f(x) from a to b. An example of this is if f(x) = x^2, an antiderivative is F(x) = (1/3)x^3. According to the theorem, the definite integral from 0 to 2 is F(2) - F(0), which gives us (1/3)(2^3) - (1/3)(0^3), ie., 8/3.