The Fundamental Theorem of Calculus is a crucial concept in calculus that connects the concepts of differentiation and integration. It has two parts: the first part states that if a function is continuous on a closed 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) - F(a).

Let's understand this concept with an example. Consider the function f(x) = 2x on the interval [0, 3]. We can find an antiderivative F(x) of f(x), which is x^2. Now, using the Fundamental Theorem of Calculus, we can evaluate the definite integral of f(x) from 0 to 3 as F(3) - F(0) = 9 - 0 = 9.

The second part of the Fundamental Theorem of Calculus provides us with a method to evaluate definite integrals. If F(x) is any antiderivative of f(x), then, for any two points a and b, the definite integral of f(x) from a to b is equal to F(b) - F(a). This means that we can bypass finding the antiderivative by simply substituting the values of a and b into F(x) and subtracting them.

Let's continue with the previous example. Instead of finding the antiderivative, we can directly evaluate the integral of f(x) from 0 to 3 using the second part of the theorem. We substitute a = 0 and b = 3 into F(x), which is x^2, and get 3^2 - 0^2 = 9.

Remember, the Fundamental Theorem of Calculus helps us relate differentiation and integration. It allows us to find definite integrals by evaluating a function at two points and subtracting the values. So let's embrace calculus with a positive attitude and conquer it together!