Definite Integrals - Complete Interactive Lesson
Part 1: Riemann Sums
โซ Riemann Sums
Part 1 of 7 โ Riemann Sums
The Area Problem
How do we find the exact area under a curve? We approximate it using rectangles, then take the limit as the number of rectangles approaches infinity.
Left, Right, and Midpoint Sums
For on with subintervals of width :
Worked Example
Approximate using a Left Riemann Sum with .
. Left endpoints: .
(The exact answer is , so is an underestimate since is increasing.)
Compute Riemann Sums ๐ฏ
Trapezoidal Rule
The trapezoidal rule averages the left and right sums:
Trapezoidal Rule from a Table ๐ฏ
Given the table:
| 0 | 2 | 5 | 8 | 10 | |
|---|---|---|---|---|---|
| 3 | 7 | 11 | 6 | 4 |
Key Takeaways โ Part 1
- Riemann Sums approximate the area under a curve using rectangles
- Left/Right/Midpoint use different sample points within each subinterval
- For increasing functions: Left underestimates, Right overestimates
- Trapezoidal Rule averages Left and Right sums for better accuracy
- More rectangles () means the sum approaches the exact integral
Part 2: Definite Integral Definition
โซ The Definite Integral
Part 2 of 7 โ Definite Integral Definition
From Riemann Sums to the Definite Integral
The definite integral is the limit of a Riemann sum as :
Part 3: Properties of Integrals
โซ Properties of Integrals
Part 3 of 7 โ Properties of Integrals
Essential Properties
| Property | Formula |
|---|---|
| Constant Multiple |
Part 4: Fundamental Theorem of Calculus โ Part 1
โซ Fundamental Theorem of Calculus โ Part 1
Part 4 of 7 โ FTC Part 1
The Big Idea
FTC Part 1 connects integration and differentiation as inverse operations:
Part 5: Fundamental Theorem of Calculus โ Part 2
โซ Fundamental Theorem of Calculus โ Part 2
Part 5 of 7 โ FTC Part 2 (Evaluation Theorem)
The Evaluation Theorem
Part 6: Mixed Integration Problems
โซ Problem-Solving Workshop
Part 6 of 7 โ Mixed Integration Problems
This workshop combines Riemann sums, FTC, properties of integrals, and applications.
AP-Style Mixed Problems ๐ฏ
More Practice ๐ฏ
Workshop Complete!
You combined all definite integral tools: Riemann sums, FTC, properties, symmetry, and applications.
Part 7: Comprehensive Review
โซ Review & Applications
Part 7 of 7 โ Comprehensive Review
Complete Integration Summary
| Concept | Key Formula |
|---|---|
| Riemann Sum | |
| Definite Integral |