6.4 The Fundamental Theorem of Calculus and Accumulation Functions

An accumulation function is a function defined by an integral with a variable upper limit, such as ${A(x)=\int_a^x f(t)\,dt}$. This means A(x) measures the total accumulated value of f(t) from the starting point a up to x. The remarkable result that connects this to derivatives is the Fundamental Theorem of Calculus, Part 1, which says that if f is continuous, then the derivative of the accumulation function is the original function: A'(x)=f(x). So integration and differentiation are inverse processes in this setting.

This theorem is one of the central ideas in calculus. It shows that when you accumulate a rate and then differentiate the total, you recover the original rate. This gives deep meaning to definite integrals beyond area alone. It also allows you to differentiate expressions involving integrals without first finding a closed-form antiderivative. The key is to recognize when a function is written as an integral with a variable endpoint and then apply the theorem directly, while being careful about the variable used inside the integral.