What is Integration?

Integration is a way of adding up quantities over a certain range or interval. It’s an important idea in calculus, especially when we want to find the area under curves or add up very tiny amounts.

Understanding Integration as Adding Up Values

1. Basic Idea:

   • At its simplest, integration helps us figure out the total amount of a function’s values. If you were to draw a graph of a function, called $f(x)$, between two points (let's call them $a$ and $b$), the integral will give you the area under that curve from $a$ to $b$.

2. Riemann Sums:

   • To understand integration better, we can start with something called Riemann sums. In this method, we break the range between $a$ and $b$ into smaller parts, called subintervals. We then look at the function’s values at certain points in these parts. We can estimate the area by adding up the areas of rectangles: $S_n = \sum_{i=1}^{n} f(x_i^*) \Delta x$
     Here, $\Delta x$ is the width of each part, and $x_i^*$ is a point we pick within each part.

3. Taking the Limit:

   • As we keep splitting the intervals into more and more parts (almost to infinity), the width of each part gets really, really small. At this point, the Riemann sum becomes the definite integral of the function: $\int_{a}^{b} f(x) \, dx = \lim_{n \to \infty} S_n$
     This process of taking the limit shows how we are continually adding up tiny areas.

How Integration is Used

• Where Integration is Applied:

  • Integration is used in many real-life situations, especially in physics, engineering, and economics. For instance, in physics, if you know the speed of something over time, you can calculate the total distance it has traveled with: $\text{Distance} = \int_{t_0}^{t_1} v(t) \, dt$

• Properties of Integrals:

  • There’s a key rule in calculus called the Fundamental Theorem of Calculus. It connects differentiation (which looks at how things change) with integration. It tells us that if $F(x)$ is a function that gives us the area under $f(x)$, then: $\int_{a}^{b} f(x) \, dx = F(b) - F(a)$
    This shows how adding up areas through integration connects with how the original function changes.

In short, integration is a powerful mathematical tool that helps us add values together. It lets us calculate areas under curves, total quantities, and more!