Welcome back, class! In Section 5.1, we approximated the area under a curve using rectangles (Riemann Sums). In Section 5.2, we take the next logical step: we make those rectangles infinitely thin to find the exact area.

The Definite Integral

We define the Definite Integral as the limit of the Riemann sums as the number of rectangles ($n$) approaches infinity. Mathematically, we express this as:

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

Here is a breakdown of the notation you need to know:

  • $\int$: The integral sign (an elongated 'S' for 'sum').
  • $a$ and $b$: The lower and upper limits of integration.
  • $f(x)$: The integrand (the function we are integrating).
  • $dx$: Indicates the variable of integration and represents the width of the infinitely thin slice.

Geometric Interpretation: Net Signed Area

One of the most important concepts in this section is Net Signed Area. When we calculate a definite integral, we aren't just looking for geometric area; we are looking for the signed difference:

$$ \int_{a}^{b} f(x) \, dx = (\text{Area above the } x\text{-axis}) - (\text{Area below the } x\text{-axis}) $$

If the function is below the $x$-axis, the integral contributes a negative value. Keep this in mind when you are evaluating integrals using geometry!

Key Properties of the Definite Integral

To evaluate integrals efficiently, we use several algebraic properties. Here are the ones we covered in the lecture video:

  1. Reversing Limits: $\int_{b}^{a} f(x) \, dx = -\int_{a}^{b} f(x) \, dx$
  2. Zero Width: $\int_{a}^{a} f(x) \, dx = 0$
  3. Constant Multiple: $\int_{a}^{b} c f(x) \, dx = c \int_{a}^{b} f(x) \, dx$
  4. Sum/Difference: $\int_{a}^{b} [f(x) \pm g(x)] \, dx = \int_{a}^{b} f(x) \, dx \pm \int_{a}^{b} g(x) \, dx$
  5. Additivity (Splitting the Interval): $\int_{a}^{b} f(x) \, dx = \int_{a}^{c} f(x) \, dx + \int_{c}^{b} f(x) \, dx$

Class Resources

I have attached the PowerPoint slides below, which contain specific examples of evaluating integrals using geometric formulas (semicircles, triangles, and rectangles). Please review these before attempting the homework.

Keep up the great work, and happy integrating!