The Fundamental Theorem of Calculus and Definite Integrals

#6.7 The Fundamental Theorem of Calculus and Definite Integrals

Welcome to one of the most important theorems in all of calculus! In this key topic, we’ll explain what the two parts of the fundamental theorem of calculus (FTOC) are as well as explain the relationship between a definite integral and its antiderivative!

#1️⃣ Fundamental Theorem of Calculus Part 1

The first part of the FTOC states the relationship between antiderivatives and definite integrals. Now what does this mean? 🤔 Let’s write this out.

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

$$g'(x)=f(x)$$

What the equation above represents is that if you take the derivative of an integral, you will be left with the inside function. This happens because differentiation and integration cancel each other out, leaving you with the inside function.

However, you need to be cautious! If the upper bound is not just $x$, then you will have to substitute whatever the upper bound is for $t$.

#✏️ Example 1 - Normal FTOC

Find $g'(x)$.

$$g(x)=\int_{32}^{x} 5t^4 dt$$

Using the FTOC, you can simplify the right side by taking the derivative of both sides. Therefore, $g'(x)=5x^4$.

#✏️ Example 2 - Upp...