Riemann Sums and Definite Integrals practice problems

📚 Topic Summary

Riemann Sums are a way to approximate the area under a curve by dividing it into rectangles. The area of each rectangle is calculated and then summed up. Definite integrals, on the other hand, give the exact area under the curve between two limits. The Fundamental Theorem of Calculus links these two concepts, showing how definite integrals can be evaluated using antiderivatives. This worksheet focuses on practicing the basic concepts and calculations associated with both.

🧠 Part A: Vocabulary

Match the terms on the left with their definitions on the right:

1. Term: Riemann Sum
2. Term: Definite Integral
3. Term: Upper Sum
4. Term: Lower Sum
5. Term: Partition

1. Definition: A sum using the maximum value of the function on each subinterval.
2. Definition: A sum that approximates the area under a curve using rectangles.
3. Definition: The exact area under a curve between two limits.
4. Definition: A sum using the minimum value of the function on each subinterval.
5. Definition: Division of an interval into subintervals.

✍️ Part B: Fill in the Blanks

Complete the following paragraph using the words provided:

The _______ is a method for approximating the area under a curve. This involves dividing the area into _________ and summing their areas. A more precise way to find the area is by using a _________. According to the Fundamental Theorem of _________, we can evaluate this if we know the _________.

Words: rectangles, Riemann Sum, antiderivative, definite integral, Calculus

🤔 Part C: Critical Thinking

Explain, in your own words, how Riemann Sums and Definite Integrals are related. What does increasing the number of rectangles in a Riemann Sum do to the accuracy of the approximation?