Understanding Definite Integrals Using the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus (FTC) is a big idea that shows how differentiation (finding slopes) and integration (finding areas) are connected. This theorem gives us a handy way to calculate definite integrals, which are important because they can be used in many areas like physics, economics, and engineering. Let’s look at how to evaluate definite integrals using the FTC, with some examples that show why this is useful in the real world.

How to Evaluate Definite Integrals with the FTC

To use the FTC for evaluating definite integrals, we will focus on the first part of the theorem. It says that if a function $f$ is continuous between two points $a$ and $b$, and if $F$ is an antiderivative (a function that gives us back the original function when we take the derivative) of $f$, then:

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

This formula helps us find the area under the curve $f(x)$ from point $x = a$ to point $x = b$.

Example 1: Finding the Area Under a Parabola

Let’s look at the definite integral:

$\int_1^3 (2x + 1)\,dx.$

1. Finding the Antiderivative: First, we find the antiderivative $F(x)$ of $f(x) = 2x + 1$. We get:

   $F(x) = x^2 + x + C,$

   where $C$ is just a constant (we can ignore it for now).

2. Evaluating at the Bounds: Next, we calculate $F(x)$ at the two endpoints:

   • For $x = 3$: $F(3) = 3^2 + 3 = 9 + 3 = 12.$

   • For $x = 1$: $F(1) = 1^2 + 1 = 1 + 1 = 2.$

3. Calculating the Definite Integral: Now, we put these values back into the formula:

$\int_1^3 (2x + 1)\,dx = F(3) - F(1) = 12 - 2 = 10.$

So, the area under the curve from $x = 1$ to $x = 3$ is 10 square units.

Example 2: Evaluating a Sine Function

Now, let's evaluate another integral:

$\int_0^{\frac{\pi}{2}} \sin(x)\,dx.$

1. Finding the Antiderivative: The antiderivative of $\sin(x)$ is:

   $F(x) = -\cos(x) + C.$

2. Evaluating at the Bounds:

   • For $x = \frac{\pi}{2}$: $F\left(\frac{\pi}{2}\right) = -\cos\left(\frac{\pi}{2}\right) = 0.$

   • For $x = 0$: $F(0) = -\cos(0) = -1.$

3. Calculating the Definite Integral:

   $\int_0^{\frac{\pi}{2}} \sin(x)\,dx = F\left(\frac{\pi}{2}\right) - F(0) = 0 - (-1) = 1.$

In this case, the area under the sine curve from $0$ to $\frac{\pi}{2}$ is 1 square unit, which can show the total distance traveled in physics.

Why Evaluating Definite Integrals Matters

Knowing how to evaluate definite integrals with the FTC is not just about math; it has important uses in the real world. For example:

• Finding Areas: In science and engineering, knowing the area under a curve can help us measure things like the area of a plot of land, how much liquid fills a tank, or trends in economic data.

• Calculating Displacement: In physics, using integrals lets us calculate how far something has moved based on its speed. This helps us find the total distance traveled over time.

When we connect these math ideas to real-life situations, it shows how evaluating definite integrals helps us understand calculus and solve everyday problems.