Test Questions: Approximating Area with Rectangles in Calculus

📚 Quick Study Guide

• 📐 Riemann Sums: Approximating the area under a curve using rectangles.
• ⬅️ Left Riemann Sum: Uses the left endpoint of each rectangle's base to determine its height.
• ➡️ Right Riemann Sum: Uses the right endpoint of each rectangle's base to determine its height.
• midpoint of each rectangle's base to determine its height.
• ➗ Formula: $\sum_{i=1}^{n} f(x_i) \Delta x$, where $\Delta x = \frac{b-a}{n}$ and $x_i$ is the chosen point within each subinterval.
• 🎯 Definite Integral: The limit of Riemann sums as the number of rectangles approaches infinity gives the exact area.
• 🤔 Overestimation/Underestimation: Depends on whether the function is increasing or decreasing and the type of Riemann sum used.

✍️ Practice Quiz

1. Question 1: Which method uses the left endpoint of each subinterval to determine the height of the rectangle in approximating the area under a curve?
   1. Left Riemann Sum
   2. Right Riemann Sum
   3. Midpoint Rule
   4. Trapezoidal Rule
2. Question 2: What does $\Delta x$ represent in the Riemann sum formula $\sum_{i=1}^{n} f(x_i) \Delta x$?
   1. The height of the rectangle
   2. The width of the rectangle
   3. The area of the rectangle
   4. The number of rectangles
3. Question 3: If a function is increasing on an interval, which Riemann sum will overestimate the area under the curve?
   1. Left Riemann Sum
   2. Right Riemann Sum
   3. Midpoint Rule
   4. All of the above
4. Question 4: What is the limit of Riemann sums as the number of rectangles approaches infinity?
   1. An approximation of the area
   2. The exact area
   3. Zero
   4. Infinity
5. Question 5: Which rule uses the average of the left and right endpoints to approximate the area under a curve?
   1. Left Riemann Sum
   2. Right Riemann Sum
   3. Midpoint Rule
   4. Trapezoidal Rule
6. Question 6: Suppose you are using the right Riemann sum with $n=4$ to approximate the area under the curve $f(x) = x^2$ from $x=0$ to $x=2$. What is the value of $\Delta x$?
   1. 0.25
   2. 0.5
   3. 1
   4. 2
7. Question 7: Which of the following is the Riemann sum formula?
   1. $\int_{a}^{b} f(x) dx$
   2. $\sum_{i=1}^{n} f(x_i) \Delta x$
   3. $f'(x)$
   4. $F(b) - F(a)$