While Riemann sums give us an approximation of the area under a curve, the definite integral gives us the exact area. It is defined as the limit of Riemann sums as the number of rectangles approaches infinity.
🎯 In this section you will learn
- The formal definition of the definite integral
- Geometric interpretation: net area (area above x-axis minus area below)
- Properties of definite integrals
- How to evaluate simple definite integrals using geometry
📌 Definition of the Definite Integral
$$ \int_a^b f(x) \, dx = \lim_{n \to \infty} \sum_{i=1}^n f(x_i^*) \Delta x $$
where $\Delta x = \frac{b-a}{n}$ and $x_i^*$ is any sample point in the $i$-th subinterval
💡 Geometric Interpretation
The definite integral represents the net area between the curve $y = f(x)$ and the x-axis from $x = a$ to $x = b$:
- Area above the x-axis counts as positive
- Area below the x-axis counts as negative
📐 Definite Integrals from Geometry
For simple shapes, we can evaluate definite integrals using area formulas:
Find $\int_1^5 3 \, dx$.
①
Interpret geometrically
$f(x)=3$ is a horizontal line. From $x=1$ to $x=5$, the region is a rectangle.
②
Area = width × height
Width = $5-1 = 4$, height = $3$ → $4 \times 3 = 12$
③
Result
$\int_1^5 3 \, dx = 12$
Find $\int_0^2 x \, dx$.
- $f(x)=x$ forms a right triangle with base $2$ and height $2$
- Area = $\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 2 \times 2 = 2$
- $\int_0^2 x \, dx = 2$
Find $\int_0^2 (-x) \, dx$.
- $f(x)=-x$ is below the x-axis. The triangle area is $2$, but it's below → negative
- $\int_0^2 (-x) \, dx = -2$
Find $\int_{-2}^2 \sqrt{4 - x^2} \, dx$.
- $y = \sqrt{4 - x^2}$ is the top half of a circle of radius $2$
- Area of full circle = $\pi r^2 = 4\pi$
- Area of semicircle = $2\pi$
- $\int_{-2}^2 \sqrt{4 - x^2} \, dx = 2\pi$
📋 Properties of Definite Integrals
| Property | Formula |
|---|
| Order of integration | $\int_a^b f(x) dx = -\int_b^a f(x) dx$ |
| Zero width | $\int_a^a f(x) dx = 0$ |
| Constant multiple | $\int_a^b c f(x) dx = c \int_a^b f(x) dx$ |
| Sum/Difference | $\int_a^b [f(x) \pm g(x)] dx = \int_a^b f(x) dx \pm \int_a^b g(x) dx$ |
| Additivity | $\int_a^b f(x) dx + \int_b^c f(x) dx = \int_a^c f(x) dx$ |
| Comparison | If $f(x) \ge g(x)$ on $[a,b]$, then $\int_a^b f(x) dx \ge \int_a^b g(x) dx$ |
If $\int_0^2 f(x) dx = 5$ and $\int_0^2 g(x) dx = 3$, find $\int_0^2 [2f(x) - g(x)] dx$.
- $\int_0^2 [2f(x) - g(x)] dx = 2\int_0^2 f(x) dx - \int_0^2 g(x) dx$
- $= 2(5) - 3 = 10 - 3 = 7$
If $\int_0^3 f(x) dx = 10$ and $\int_3^5 f(x) dx = 4$, find $\int_0^5 f(x) dx$.
- $\int_0^5 f(x) dx = \int_0^3 f(x) dx + \int_3^5 f(x) dx = 10 + 4 = 14$
⚠️ Important Notes
- The definite integral is a number (not a function). It does not include $+C$.
- Area below the x-axis contributes negative area to the definite integral.
- The definite integral can be interpreted as "net area" (total area above minus total area below).
- Use geometry to evaluate definite integrals when $f(x)$ is a simple shape (rectangle, triangle, circle, etc.).
🔍 Key Takeaways
- Definite integral: $\int_a^b f(x) dx = \lim_{n \to \infty} \sum f(x_i^*) \Delta x$
- Geometrically: net area between the curve and the x-axis.
- Above x-axis = positive area, below x-axis = negative area.
- Properties include constant multiple, sum/difference, additivity, and order reversal.
- Simple definite integrals can be evaluated using geometric formulas (rectangle, triangle, circle).
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