If we can find the antiderivative function F ( x ) of the integrand f ( x ) , then the definite integral ∫ b a f ( x ) d x can be determined by F ( b ) − F ( a ) provided that f ( x ) is continuous. We are usually given continuous functions, but if you want to be rigorous in your solutions, you should state that f ( x ) is continuous and why. FTC part 2 is a very powerful statement. Recall in the previous chapters, the definite integral was calculated from areas under the curve using Riemann sums. FTC part 2 just throws that all away. We just have to find the antiderivative and evaluate at the bounds! This is a lot less work. For most students, the proof does give any intuition of why this works or is true. But let's look at s ( t ) = ∫ b a v ( t ) d t . We know that integrating the velocity function gives us a position function. So taking s ( b ) − s ( a ) results in a displacement.
