📚 Topic Summary
Definite integrals represent the signed area under a curve between two points on the x-axis. Riemann Sums are a method of approximating this area by dividing it into rectangles and summing their areas. By taking the limit as the width of these rectangles approaches zero, we can find the exact value of the definite integral. This worksheet will help you understand and apply the concept of using Riemann Sums to evaluate definite integrals.
🧠 Part A: Vocabulary
Match the following terms with their definitions:
| Term |
Definition |
| 1. Definite Integral |
A. The width of each subinterval in a Riemann Sum. |
| 2. Riemann Sum |
B. The limit of a sum of areas of rectangles used to approximate the area under a curve. |
| 3. Partition |
C. A method of approximating the definite integral using rectangles. |
| 4. Subinterval |
D. The area under a curve between two specified limits. |
| 5. $\Delta x$ |
E. A division of the interval $[a, b]$ into smaller intervals. |
✏️ Part B: Fill in the Blanks
Complete the following paragraph with the missing words:
A Riemann Sum is an approximation of the ________ ________. It is calculated by dividing the interval into ________ and forming ________. The area of each rectangle is found and then ________ together. As the width of each subinterval approaches ________, the Riemann Sum approaches the exact value of the definite integral.
🤔 Part C: Critical Thinking
Explain why the Riemann Sum is an approximation of the definite integral and not the exact value in most cases. Under what conditions would a Riemann Sum give the exact value of the definite integral?