Definite Integral examples using the limit definition

📚 Quick Study Guide

• 🔍 The definite integral of a function $f(x)$ from $a$ to $b$ is denoted as $\int_{a}^{b} f(x) dx$.
• 🔢 The limit definition of the definite integral is: $\int_{a}^{b} f(x) dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i) \Delta x$, where $\Delta x = \frac{b-a}{n}$ and $x_i = a + i\Delta x$.
• 💡 Common summation formulas:
  • $\sum_{i=1}^{n} i = \frac{n(n+1)}{2}$
  • $\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}$
  • $\sum_{i=1}^{n} i^3 = (\frac{n(n+1)}{2})^2$
  • $\sum_{i=1}^{n} c = nc$, where $c$ is a constant.
• 📝 Remember to simplify the expression inside the summation before applying the limit.
• ➗ Be careful with algebraic manipulations and fractions.
• ➕ Don't forget to distribute and combine like terms.

🧪 Practice Quiz

1. Question 1: Evaluate $\int_{0}^{2} x dx$ using the limit definition.
1. 0
2. 2
3. 4
4. 6
2. Question 2: Evaluate $\int_{1}^{3} (2x+1) dx$ using the limit definition.
1. 4
2. 8
3. 12
4. 16
3. Question 3: Evaluate $\int_{0}^{1} x^2 dx$ using the limit definition.
1. $\frac{1}{2}$
2. $\frac{1}{3}$
3. $\frac{1}{4}$
4. $\frac{1}{6}$
4. Question 4: Evaluate $\int_{0}^{2} (x^2 + 1) dx$ using the limit definition.
1. $\frac{2}{3}$
2. $\frac{8}{3}$
3. $\frac{14}{3}$
4. $\frac{16}{3}$
5. Question 5: Evaluate $\int_{1}^{2} x^3 dx$ using the limit definition.
1. $\frac{15}{4}$
2. $\frac{17}{4}$
3. $\frac{19}{4}$
4. $\frac{21}{4}$
6. Question 6: Evaluate $\int_{0}^{1} (3x^2 + x) dx$ using the limit definition.
1. $\frac{3}{2}$
2. $\frac{5}{2}$
3. 1
4. 2
7. Question 7: Evaluate $\int_{-1}^{1} x dx$ using the limit definition.
1. -2
2. 0
3. 2
4. 1