The first part of the Fundamental Theorem of Calculus is a powerful tool in evaluating definite integrals and finding the area under curves. Let's explore some examples to understand its applications.

Example 1: Suppose we have the function f(x) = 2x^2 + 3. We want to find the area under the curve of f(x) between x = 1 and x = 4. According to the first part of the Fundamental Theorem, we can evaluate this definite integral by finding the antiderivative of f(x) and then subtracting the values at the endpoints. In this case, the antiderivative of f(x) is F(x) = (2/3)x^3 + 3x. Substituting the values, we get F(4) - F(1) = (2/3)(4^3) + 3(4) - [(2/3)(1^3) + 3(1)]. Simplifying this expression gives us the desired area under the curve.

Example 2: Consider the function g(x) = 1/x. To find the area under the curve of g(x) between x = 2 and x = 5, we need to evaluate the definite integral of g(x). The antiderivative of g(x) is ln|x|, so the area under the curve is ln(5) - ln(2).

Example 3: Let's take a look at a more challenging example. Suppose we have the function h(x) = 2cos(x). To find the area under the curve of h(x) between x = 0 and x = π/2, we can evaluate the definite integral by finding the antiderivative of h(x), which is 2sin(x). Substituting the values, we get 2sin(π/2) - 2sin(0) = 2 - 0 = 2.

By applying the first part of the Fundamental Theorem of Calculus, we can easily evaluate definite integrals and find the area under curves. It provides us with a direct relationship between differentiation and integration, making complex calculations more accessible.

Remember, practice is key to mastering any mathematical concept. So keep exploring and solving problems using the first part of the Fundamental Theorem, and soon you'll become a calculus pro!