Integration and Its Applications

Introduction to Integration

Integration is a fundamental concept in calculus, used to find areas under curves, accumulate quantities, and solve problems involving rates of change. This study guide covers the basics of definite integrals, Riemann sums, and the Fundamental Theorem of Calculus, along with practical applications and sample problems.

• Definite Integral: Represents the signed area under a curve between two points. Denoted as .

• Riemann Sum: An approximation of the area under a curve using a sum of areas of rectangles.

• Fundamental Theorem of Calculus: Connects differentiation and integration, providing a method to evaluate definite integrals.

• Use velocity data from a table to estimate change in position.

Allowed Resources During Exams

• Basic four-function or scientific calculator (no graphing calculators).

• No cell phones, computers, or other books/notes allowed.

Key Exam Topics

• Definite Integrals: Understanding and computing .

• Riemann Sums: Approximating integrals using left, right, or midpoint rectangles.

• Properties of Integrals: Linearity, additivity, and the effect of reversing limits.

• Geometric Interpretation: Using geometry to compute areas under simple curves.

• Fundamental Theorem of Calculus (FTC): If , then .

Definite Integrals and Riemann Sums

Defining the Definite Integral

The definite integral of a function from to is defined as the limit of a Riemann sum as the number of subintervals approaches infinity:

• Where and is a sample point in the th subinterval.

Left and Right Riemann Sums

• Left Riemann Sum: Uses the left endpoint of each subinterval for the height of the rectangle.

• Right Riemann Sum: Uses the right endpoint of each subinterval for the height.

• Formula for Right Riemann Sum: where

Example: Approximating an Integral

• Given on , approximate using a left Riemann sum with 3 rectangles.

• Draw rectangles, compute their areas, and sum for the approximation.

Geometric Interpretation of Integrals

For linear or simple functions, the area under the curve can often be found using basic geometry (e.g., area of a triangle or rectangle).

• Example: The area under from to forms a triangle. Use .

Properties of Definite Integrals

• Linearity:

• Additivity:

• Reversal of Limits:

Fundamental Theorem of Calculus (FTC)

Statement and Application

• Part 1: If , then .

• Part 2: , where is any antiderivative of .

Example:

• Given , then by FTC Part 1.

Applications: Velocity and Accumulation

Integrals can be used to compute accumulated change, such as distance traveled given a velocity function.

• Distance from Velocity:

• Analyze intervals where is increasing (velocity positive) or decreasing (velocity negative).

Sample Practice Problems

• Approximate using left and right Riemann sums.

• Use geometry to find the exact value of .

• Given a velocity graph, compute and interpret the result as net distance.

• Given , find .

• Evaluate definite integrals such as .

Table: Properties of Definite Integrals

Summary

• Understand how to compute and interpret definite integrals.

• Be able to approximate integrals using Riemann sums and geometric reasoning.

• Apply the Fundamental Theorem of Calculus to evaluate and differentiate integrals.

• Practice with sample problems to reinforce understanding and exam readiness.