Welcome back to Professor Baker's Math Class! In this lesson, we bridge the gap between geometry and calculus by introducing Integration Basics. We start with a simple question: We know how to find the area of a rectangle ($A=lw$) or a triangle ($A=\frac{1}{2}bh$), but how do we find the area under a curved line?

1. From Rectangles to Riemann Sums

To find the area under a curve like $y=x^2$, we can approximate it by drawing rectangles underneath the graph. As we saw in the class notes, using just a few rectangles gives us a rough estimate. However, if we increase the number of rectangles ($n$) to infinity, the width of each rectangle gets infinitely small, and our approximation becomes exact.

This limit is the definition of the Definite Integral:

$$ \int_{a}^{b} f(x) \, dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i) \Delta x $$

2. Net Signed Area

One of the most important concepts in this section is that the definite integral represents the net signed area. This means:

  • Area above the x-axis is positive.
  • Area below the x-axis is negative.

We looked at the example $\int_{0}^{3} (x-1) \, dx$. Geometrically, this graph creates a small triangle below the axis (negative area) and a larger triangle above the axis (positive area). When we integrate, we subtract the area below from the area above to get the final answer of $\frac{3}{2}$ or $1.5$.

3. Properties of Definite Integrals

Integration follows specific algebraic rules that make calculations easier. Here are the key properties we discussed:

  1. Sum Rule: $\int [f(x) + g(x)] \, dx = \int f(x) \, dx + \int g(x) \, dx$
  2. Difference Rule: $\int [f(x) - g(x)] \, dx = \int f(x) \, dx - \int g(x) \, dx$
  3. Constant Multiple Rule: $\int c \cdot f(x) \, dx = c \cdot \int f(x) \, dx$

4. Evaluating Integrals: Examples

Using the Fundamental Theorem of Calculus, we can evaluate integrals by finding the antiderivative. Let's look at two examples from our notes:

Example A: Polynomials
To solve $\int_{0}^{1} (4 + 3x^2) \, dx$:

  1. Find the antiderivative: $4x + x^3$.
  2. Evaluate at the bounds $1$ and $0$: $$ (4(1) + (1)^3) - (4(0) + (0)^3) = 5 - 0 = 5 $$

Example B: The Exponential Function
We also looked at $\int_{1}^{3} e^x \, dx$. Since the derivative of $e^x$ is just $e^x$, the antiderivative is also $e^x$.

$$ \left[ e^x \right]_{1}^{3} = e^3 - e^1 \approx 17.4 $$

5. Practice Makes Perfect

We wrapped up with a more complex polynomial: $\int_{-3}^{2} (x^2 - 2x + 3) \, dx$. By applying the power rule to each term—$\frac{1}{3}x^3 - x^2 + 3x$—and carefully plugging in our upper and lower limits, we found the total area to be $95/3$.

Keep practicing these properties and remember to watch your signs when calculating areas below the x-axis!