The Fundamental Theorem of Calculus (FTC) is super important for finding areas under curves.

In simple terms, it links two big ideas in math: differentiation and integration. This link makes it easier to calculate how much space is below a curve on a graph.

What is FTC?

The FTC has two main parts:

1. First Part - If you have a smooth function called $f(x)$, and you want to find the area $A$ between two points, $a$ and $b$, you can write it as: $A = \int_{a}^{b} f(x) \, dx.$

2. Second Part - If you take the area function and find its derivative, you will get back the original function: $\frac{d}{dx} \left( \int_{a}^{x} f(t) \, dt \right) = f(x).$

How to Find Areas

Let’s look at an example. Suppose you want to find the area under the curve of $f(x) = x^2$ from $x = 1$ to $x = 3$.

You would set up the integral like this: $A = \int_{1}^{3} x^2 \, dx.$

By using the power rule of integration, you can calculate the area: $A = \left[ \frac{x^3}{3} \right]_{1}^{3} = \frac{27}{3} - \frac{1}{3} = \frac{26}{3}.$

This area shows how much space is between the curve $y = x^2$, the x-axis, and the vertical lines at $x = 1$ and $x = 3$.

The FTC makes finding areas easier and helps us understand how things add up, which is a key part of calculus!