#Definite Integrals Study Guide

#Table of Contents

1. Introduction to Definite Integrals
2. Definite Integral as a Limit of Riemann Sums
3. Properties of Definite Integrals
4. Practice Questions
5. Glossary
6. Summary and Key Takeaways

#Introduction to Definite Integrals

#What is a Definite Integral?

A definite integral is written as:

$$\int_{a}^{b} f(x) , dx$$

• Integrand ($f(x)$): The function being integrated.
• Integration Variable ($dx$): Indicates the variable of integration.
• Limits of Integration ($a$ and $b$): The bounds between which the function is integrated.
  • The function is integrated 'from $a$ to $b$'.
  • Typically, $a \leq b$, but $a > b$ is also valid.

**Example**: If $f(x) \geq 0$ on the interval $[a, b]$, the definite integral $\int\_{a}^{b} f(x) \, dx$ gives the area between the function $f(x)$ and the $x$-axis, from $x = a$ to $x = b$.

#Interpreting Definite Integrals

• A definite integral outputs a number based on $f(x)$ and the values of $a$ and $b$.
• If $f(x)$ is a rate of change function, the definite integral represents the accumulation of change from $x = a$ to $x = b$.

**Note**: The definite integral can also be interpreted as the signed area under the curve, where areas above the $x$-axis are positive and those below are negative.

#Definite Integral as a Limit of Riemann Sums

#Definition

The value of a definite integral can be defined as a limit of Riemann sums:

$$\int_{a}^{b} f(x) , dx = \lim_{{\Delta x_i \to 0}} \sum_{i=1}^{n} f(x_i^*) \Delta x_i$$

• $n$: The number of subintervals.
• $x_i^*$: Any value within the $i$-th subinterval.
• $\Delta x_i$: The width of the $i$-th subinterval.
• $\lim_{{\Delta x_i \to 0}}$: The limit as the width of the largest subinterval approaches zero.

**Example**: As the subintervals get narrower, the area of the approximating rectangles gets closer to the exact area under the curve.

#Alternate Form

The Riemann sum limit for a definite integral can also be written as:

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

If the subintervals are of equal width:

$$\Delta x = \frac{b - a}{n}$$

#Worked Example

Evaluate the following limit:

$$\lim_{{n \to \infty}} \sum_{i=1}^{n} \left(-3 + \frac{5}{n}i\right)^2 + 1 \cdot \left(\frac{5}{n}\right)$$

Solution:

Recognize this as a limit of Riemann sums. The key term is $\frac{5}{n}$, indicating the width of subintervals. The interval is $[-3, 2]$.

$$\int_{-3}^{2} (x^2 + 1) , dx$$

Evaluate the integral:

$$\int_{-3}^{2} (x^2 + 1) , dx = \left[ \frac{x^3}{3} + x \right]_{-3}^{2} = \frac{14}{3} - (-12) = \frac{50}{3}$$

Thus,

$$\lim_{{n \to \infty}} \sum_{i=1}^{n} \left(-3 + \frac{5}{n}i\right)^2 + 1 \cdot \left(\frac{5}{n}\right) = \frac{50}{3}$$

#Properties of Definite Integrals

#Definite Integral of a Constant Times a Function

If $k$ is a constant:

$$\int_{a}^{b} k f(x) , dx = k \int_{a}^{b} f(x) , dx$$

#Definite Integral of a Sum or Difference of Functions

If $f$ and $g$ are two functions:

$$\int_{a}^{b} (f(x) \pm g(x)) , dx = \int_{a}^{b} f(x) , dx \pm \int_{a}^{b} g(x) , dx$$

**Example**: $\int\_{0}^{2} (3x + 4) \, dx = \int\_{0}^{2} 3x \, dx + \int\_{0}^{2} 4 \, dx$

#Changing Limits of Integration

#Zero-Length Interval

For any function $f$:

$$\int_{a}^{a} f(x) , dx = 0$$

#Reversing the Limits

For a function $f$:

$$\int_{b}^{a} f(x) , dx = -\int_{a}^{b} f(x) , dx$$

#Adjacent Intervals

For a function $f$, and $c$ such that $a \leq c \leq b$:

$$\int_{a}^{b} f(x) , dx = \int_{a}^{c} f(x) , dx + \int_{c}^{b} f(x) , dx$$

**Example**: If $\int\_{2}^{5} f(x) \, dx = -3$ and $\int\_{2}^{7} f(x) \, dx = 12$, find $\int\_{5}^{7} f(x) \, dx$.

$$\int_{2}^{7} f(x) , dx = \int_{2}^{5} f(x) , dx + \int_{5}^{7} f(x) , dx$$

$$12 = -3 + \int_{5}^{7} f(x) , dx$$

$$\int_{5}^{7} f(x) , dx = 15$$

#Practice Questions

Practice Question

Question 1: Evaluate $\int_{0}^{3} (2x^2 + 3x) , dx$.

Answer: $$\int_{0}^{3} (2x^2 + 3x) , dx = \left[ \frac{2x^3}{3} + \frac{3x^2}{2} \right]_{0}^{3} = \left(18 + 13.5\right) - 0 = 31.5$$

Practice Question

Question 2: If $\int_{1}^{4} f(x) , dx = 7$ and $\int_{4}^{6} f(x) , dx = -2$, find $\int_{1}^{6} f(x) , dx$.

Answer: $$\int_{1}^{6} f(x) , dx = \int_{1}^{4} f(x) , dx + \int_{4}^{6} f(x) , dx = 7 + (-2) = 5$$

Practice Question

Question 3: Use the property of reversing limits to evaluate $\int_{3}^{1} 5x , dx$.

Answer: $$\int_{3}^{1} 5x , dx = -\int_{1}^{3} 5x , dx = -\left[ \frac{5x^2}{2} \right]_{1}^{3} = -\left( \frac{45}{2} - \frac{5}{2} \right) = -20$$

#Glossary

• Definite Integral: The integral of a function over a specific interval, resulting in a number.
• Integrand: The function being integrated.
• Limits of Integration: The bounds $a$ and $b$ in the integral $\int_{a}^{b} f(x) , dx$.
• Riemann Sum: A method for approximating the definite integral of a function.
• Subinterval: A division of the interval $[a, b]$ used in Riemann sums.

#Summary and Key Takeaways

#Summary

• Definite integrals are used to calculate the total accumulation of a function over an interval.
• They can be interpreted as the area under the curve of the function.
• Definite integrals can be defined using Riemann sums.
• There are several properties of definite integrals, such as dealing with constants, sums, and differences.

#Key Takeaways

• Understand the notation and basic interpretation of definite integrals.
• Recognize that definite integrals can be evaluated using standard integration techniques, not Riemann sums.
• Be familiar with properties such as changing limits, reversing limits, and dealing with sums/differences of functions.

Exam Tip

Exam Tip: Practice evaluating definite integrals and understanding their geometric and physical interpretations.

Exam Tip

IB Objective: Ensure you can link definite integrals to real-world applications and problems, as well as solve them efficiently.

**Note**: Always double-check your integration limits and the function you are integrating to avoid common mistakes.