Properties of Definite Integrals: Rules and Practical Examples

📚 Quick Study Guide

• 🔢 Linearity: $\int_{a}^{b} [cf(x) + dg(x)] dx = c\int_{a}^{b} f(x) dx + d\int_{a}^{b} g(x) dx$, where $c$ and $d$ are constants. This means you can split integrals and pull out constants.
• 🔄 Reversal of Limits: $\int_{a}^{b} f(x) dx = -\int_{b}^{a} f(x) dx$. Swapping the limits of integration changes the sign of the integral.
• ➕ Additivity: $\int_{a}^{c} f(x) dx + \int_{c}^{b} f(x) dx = \int_{a}^{b} f(x) dx$. You can split an integral at any point within the interval.
• ⚖️ Symmetry (Even Function): If $f(x)$ is even (i.e., $f(-x) = f(x)$), then $\int_{-a}^{a} f(x) dx = 2\int_{0}^{a} f(x) dx$.
• 🎢 Symmetry (Odd Function): If $f(x)$ is odd (i.e., $f(-x) = -f(x)$), then $\int_{-a}^{a} f(x) dx = 0$.
• 🎯 Constant Function: $\int_{a}^{b} c \, dx = c(b-a)$, where $c$ is a constant.
• 📝 Domination: If $f(x) \ge 0$ on $[a, b]$, then $\int_{a}^{b} f(x) dx \ge 0$. If $f(x) \ge g(x)$ on $[a, b]$, then $\int_{a}^{b} f(x) dx \ge \int_{a}^{b} g(x) dx$.

🧪 Practice Quiz

1. What is the value of $\int_{2}^{2} x^2 dx$?
   1. 0
   2. 4
   3. 8
   4. 16
2. If $\int_{0}^{5} f(x) dx = 10$, what is the value of $\int_{5}^{0} f(x) dx$?
   1. 10
   2. -10
   3. 0
   4. 20
3. Given $\int_{0}^{2} x dx = 2$ and $\int_{0}^{2} x^2 dx = \frac{8}{3}$, what is the value of $\int_{0}^{2} (x + x^2) dx$?
   1. $\frac{14}{3}$
   2. $\frac{8}{3}$
   3. 2
   4. $\frac{10}{3}$
4. If $f(x)$ is an even function and $\int_{-2}^{2} f(x) dx = 6$, what is the value of $\int_{0}^{2} f(x) dx$?
   1. 3
   2. 6
   3. 0
   4. 12
5. If $f(x)$ is an odd function, what is the value of $\int_{-5}^{5} f(x) dx$?
   1. 5
   2. -5
   3. 0
   4. 10
6. What is the value of $\int_{1}^{3} 5 dx$?
   1. 5
   2. 10
   3. 15
   4. 20
7. Given $\int_{0}^{3} f(x) dx = 5$ and $\int_{3}^{6} f(x) dx = 2$, what is the value of $\int_{0}^{6} f(x) dx$?
   1. 2
   2. 3
   3. 5
   4. 7