Lesson 1.5: Calculus Review - Integrals (The "Sum")

Welcome! In our last lesson, we learned about derivatives (df/dt), which are the 'speed' of our function. Now, we'll learn about their 'opposite': integrals.

If a derivative is for "slicing" a function to find its instantaneous speed, an integral is for "summing" all those tiny slices back together to find a total amount.

The "Why": The Summing Problem

In finance, we constantly need to find a "total" from a rate that changes over time.

  • If you have a changing interest rate, what is your total interest earned over one year?
  • If you have a changing risk exposure, what is your total risk cost over one month?

You can't just do Rate × Time because the rate itself is changing. You need to...

  1. ...slice time into a million tiny moments.
  2. ...find the tiny amount of interest/cost in each tiny moment.
  3. ...sum all those million tiny pieces together.

This "summing" process is called integration.

The Core Concept: The "Riemann Sum"

The easiest way to understand an integral is to visualize it as "finding the area under a curve." Let's say we have a function f(x)f(x) and we want to find the total area under it from point aa to point bb. It's a hard problem because the top f(x)f(x) is curvy. We can't just multiply width × height.

The Solution (The "Sum"):

We can approximate this area by "slicing" it into a bunch of simple, skinny rectangles.

Interactive Riemann Sum
Adjust the slider to change the number of rectangles used to approximate the area under the curve.
Approximate Area: 34.544
  • Slice: We cut the width (from aa to bb) into nn tiny steps, each with a width of Δx\Delta x.
  • Calculate: For each slice, we draw a rectangle. Its width is Δx\Delta x and its height is the function's value, f(x)f(x). The area of one tiny rectangle is f(x)⋅Δxf(x) \cdot \Delta x.
  • Sum: We find the total area by summing up all the tiny rectangles. This is called a Riemann Sum.
Total Area≈∑if(xi)Δx\text{Total Area} \approx \sum_{i} f(x_i) \Delta x

(The ∑\sum symbol is the Greek letter "Sigma," which just means "sum.")

The "Calculus" Leap: Making the Sum Perfect

Our approximation (the sum of rectangles) has small errors, because of the "blocky" tops. How do we make the approximation perfect? We make the rectangles infinitely skinny. We let the number of slices nn go to infinity, which means our step size Δx\Delta x becomes "infinitesimally small."

When we do this, our notation changes:

  • The "tiny step" Δx\Delta x becomes the infinitesimal dxdx.
  • The "Sum" symbol ∑\sum (Sigma) gets stretched into the Integral symbol ∫\int.

This gives us the Definite Integral:

Total Area=∫abf(x)dx\text{Total Area} = \int_a^b f(x) dx

This formula literally means: "the perfect sum (∫\int) of all the tiny rectangle heights (f(x)f(x)) times their infinitely skinny widths (dxdx), from aa to bb."

What's Next? (The 'Hook')

    This "normal" integral is the master tool for summing up predictable changes. But in stochastic calculus, we will need to answer a much harder question:

    "How do we sum up random changes?"

    This will lead us to our Itô Integral (in Module 2), which looks like this:

    ∫abf(t)dWt\int_a^b f(t) dW_t

    This formula means "the sum of a function (f(t)f(t)) times a tiny random step (dWtdW_t)." Before we can build this "random" integral, we must first define our random path WtW_t. That is the goal of our next lesson.

Up Next: Taylor Expansion (Single Variable)