The Fundamental Theorem of Calculus and Accumulation Functions

#6.4 The Fundamental Theorem of Calculus and Accumulation Functions

Now that we have introduced integrals, you may be wondering how they connect with derivatives. Today, we’ll introduce the Fundamental Theorem of Calculus which connects these two topics.

#∫ Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects differentiation and integration. It states that the derivative of an integral is just the function inside the integral. 😊

We can define a new function, $F$, that represents the antiderivative of $f$.

$$F(x)=\int_{a}^{x}f(t)dt$$

The Fundamental Theorem of Calculus states that if $f$ is continuous on the interval $(a,b)$ then for every $x$ in the interval:

$$\frac{d}{dx}\left[\int_{a}^{x}f(t)dt\right]=F'(x)=f(x)$$

This means that we can use the Fundamental Theorem of Calculus to find derivatives. Let’s walk through an example to see what this means.

#✏️ Fundamental Theorem of Calculus Example

Find $g'(16)$ if $g(x)$ is the function below.

$$g(x)=\int_{5}^{x}\sqrt[4]{t}dt$$

How should we proceed? If we ignore what $g(x)$ is specifically defined as for a minute and think about how we would proceed for a function that is not defined with integrals, we would do so by first finding the der...