The Fundamental Theorem of Calculus (FTC) is really important because it connects two big ideas in math: differentiation and integration. It helps us figure out areas under curves in an easy way.

The theorem has two main parts:

Part 1: If we have a function called $f$ that is continuous between two points $a$ and $b$, and if $F$ is an antiderivative of $f$, then we can write this:

$$\int_a^b f(x) \, dx = F(b) - F(a).$$

This means that when we integrate, we’re calculating the total area under the curve of $f(x)$ from $a$ to $b$ by looking at $F$. This calculation includes both the positive and negative areas, showing how integration is all about finding area.

Part 2: This part says that if $f$ can be integrated on the interval $[a, b]$, and we define $F$ like this:

$$F(x) = \int_a^x f(t) \, dt,$$

then $F$ can be differentiated on the interval $(a, b)$. This means $F'(x) = f(x)$. So, when we find the slope of the area function $F$ at any point $x$, we get back the original function $f(x)$.

Applications in Calculus

Thanks to the FTC, we can do some cool things in calculus, like:

• Finding areas under curves: We use definite integrals for this.
• Calculating volumes of 3D shapes: For example, using techniques like the disk and washer methods.
• Finding the average value of functions: We can use this formula:

$$\text{Average value} = \frac{1}{b-a} \int_a^b f(x) \, dx.$$

In short, the FTC brings together the ideas of differentiation and integration. It gives us powerful tools to calculate areas and do even more in calculus.