singh.brandi62
singh.brandi62 May 26, 2026 • 0 views

Properties of Definite Integrals: Rules and Practical Examples

Hey there! 👋 Let's ace those definite integrals with a quick study guide and practice quiz! It's all about understanding the rules and applying them. Ready to level up your math skills? 🤓
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martinez.jamie34 Jan 7, 2026

📚 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
Click to see Answers
  1. A
  2. B
  3. A
  4. A
  5. C
  6. B
  7. D

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