Definite integrals are an important idea in AP Calculus AB. They help us find the area under curves.

When we talk about a definite integral, we write it like this:

$\int_{a}^{b} f(x) \, dx$

This means we are looking at the function ( f(x) ) from point ( a ) to point ( b ). Knowing how to use definite integrals is important for solving different real-world problems that involve areas and more.

Properties of Definite Integrals

Here are some key properties of definite integrals:

1. Additivity: You can break an integral into smaller parts:

   $$\int_{a}^{c} f(x) \, dx = \int_{a}^{b} f(x) \, dx + \int_{b}^{c} f(x) \, dx$$

   This means you can find the area under a curve in pieces. It makes it easier to work with complex shapes.

2. Reversal of Limits: If you switch the limits ( a ) and ( b ), the integral changes sign:

   $$\int_{b}^{a} f(x) \, dx = -\int_{a}^{b} f(x) \, dx$$

   This shows us that areas below the x-axis count as negative.

3. Constant Multiple: If you have a number multiplied by the function, you can take it out of the integral:

   $$\int_{a}^{b} k f(x) \, dx = k \int_{a}^{b} f(x) \, dx$$

   This helps when you need to scale areas if a function is multiplied by a constant.

4. Non-negativity: If ( f(x) ) is zero or positive in the interval ([a, b]), then the integral ( \int_{a}^{b} f(x) , dx ) gives the area above the x-axis. If ( f(x) ) is negative, the area below the x-axis will be counted as negative.

Calculating Area Under Curves

To find the area under a curve, we can use something called the Fundamental Theorem of Calculus. This connects two important ideas: differentiation and integration. The theorem says:

• If ( F(x) ) is an antiderivative (the reverse of a derivative) of ( f(x) ) from ( a ) to ( b ), then: $$\int_{a}^{b} f(x) \, dx = F(b) - F(a)$$

This helps us calculate areas quickly once we know the antiderivative of the function.

Real-World Applications

Definite integrals are used in many areas, including:

• Physics: To find displacement, velocity, and work done by a force.
• Economics: To calculate consumer and producer surplus by finding the area between supply and demand curves.
• Biology: To estimate populations or the area that a biological population covers over time.

Example Calculation

Let’s say we want to find the area under the curve ( f(x) = x^2 ) from ( x = 1 ) to ( x = 3 ). Here’s how we do it:

1. First, find the antiderivative: ( F(x) = \frac{1}{3}x^3 ).
2. Then, use the Fundamental Theorem: $$\int_{1}^{3} x^2 \, dx = F(3) - F(1) = \left(\frac{1}{3}(3)^3\right) - \left(\frac{1}{3}(1)^3\right) = 9 - \frac{1}{3} = \frac{26}{3}$$

So the area under the curve is ( \frac{26}{3} ) square units.

In summary, definite integrals are a key tool in AP Calculus AB. They help us calculate areas under curves, which is useful in many fields!