## AP Calc AB Unit 5 Review

### Introduction

As the AP Calculus AB exam approaches, it is essential to review all the topics covered in the course. Unit 5 focuses on the concept of integration and its applications. In this review, we will dive into the key concepts and skills you need to master in Unit 5 to ace the exam. Let's get started!

### Understanding Integration

Integration is the process of finding the integral of a function. It is the reverse operation of differentiation and allows us to find the area under a curve, the accumulation of quantities over time, and much more. Here are the main topics covered in Unit 5:

### Antiderivatives and Indefinite Integrals

- Antiderivatives: An antiderivative of a function f(x) is a function F(x) whose derivative is equal to f(x). It is denoted as ∫f(x)dx.

- Indefinite Integrals: An indefinite integral represents a family of functions that differ by a constant. It is denoted as ∫f(x)dx + C, where C is the constant of integration.

### The Fundamental Theorem of Calculus

- The Fundamental Theorem of Calculus states that if a function f(x) is continuous on an interval [a, b] and F(x) is any antiderivative of f(x) on that interval, then:

∫[a, b]f(x)dx = F(b) - F(a)

### Definite Integrals

- Definite Integrals: A definite integral represents the accumulation of quantities over a specific interval. It is denoted as ∫[a, b]f(x)dx.

- Properties of Definite Integrals: The definite integral has various properties, including linearity, additivity, and the average value of a function.

### Techniques of Integration

- Substitution: Substitution is a powerful technique in integration that involves replacing variables with new ones to simplify the integral.

- Integration by Parts: Integration by parts is used when integrating the product of two functions. It is based on the product rule of differentiation.

- Partial Fractions: Partial fractions are used to break down a complex rational function into simpler fractions.

- Trigonometric Substitution: Trigonometric substitution is used when dealing with integrals involving square roots of quadratic expressions.

### Applications of Integration

- Area Under a Curve: Integration allows us to find the area enclosed between a curve and the x-axis or between two curves.

- Accumulation of Quantities: Integration can be used to calculate the total accumulation of quantities over time, such as the total distance traveled or the total amount of water in a tank.

- Average Value of a Function: Integration can be used to find the average value of a function over a specific interval.

### Differential Equations

- Differential Equations: Differential equations involve the relationship between a function and its derivatives. Integration is often used to solve differential equations.

- Separable Differential Equations: Separable differential equations can be solved by separating the variables and integrating each side.

- Slope Fields: Slope fields are graphical representations of differential equations that help visualize the behavior of solutions.

### Review Questions

To solidify your understanding of Unit 5, here are some review questions for you to tackle:

1. Find the antiderivative of f(x) = 3x^2 + 2x - 5.

2. Evaluate the definite integral ∫[0, 4] (2x + 1) dx.

3. Use the Fundamental Theorem of Calculus to evaluate ∫[1, 5] (4x^3 - 2x) dx.

4. Find the integral ∫ e^x sin(x) dx using integration by parts.

5. Calculate the area enclosed between the curve y = x^2 and the x-axis from x = 0 to x = 2.

### Conclusion

Unit 5 of AP Calculus AB covers integration and its applications. By understanding the key concepts and techniques in this unit, you will be well-prepared for the AP exam. Make sure to practice solving integration problems and review the fundamental theorems to boost your confidence. Good luck!