Riemann Sums calculation examples (LRAM, RRAM, MRAM)

📚 Quick Study Guide

🔍 Riemann Sums are used to approximate the definite integral of a function.

💡 Left Riemann Sum (LRAM): Uses the left endpoint of each subinterval to determine the height of the rectangle. Formula: $LRAM = \Delta x [f(x_0) + f(x_1) + ... + f(x_{n-1})]$, where $\Delta x = \frac{b-a}{n}$.

📝 Right Riemann Sum (RRAM): Uses the right endpoint of each subinterval. Formula: $RRAM = \Delta x [f(x_1) + f(x_2) + ... + f(x_n)]$.

📊 Midpoint Riemann Sum (MRAM): Uses the midpoint of each subinterval. Formula: $MRAM = \Delta x [f(\frac{x_0+x_1}{2}) + f(\frac{x_1+x_2}{2}) + ... + f(\frac{x_{n-1}+x_n}{2})]$.

➕ $\Delta x$ represents the width of each subinterval, calculated as $\frac{b-a}{n}$, where $a$ and $b$ are the limits of integration and $n$ is the number of subintervals.

📐 A larger $n$ (more subintervals) generally leads to a more accurate approximation.

Practice Quiz

1. What does LRAM stand for?
1. A) Left Root Area Method
2. B) Left Riemann Area Method
3. C) Left Riemann Approximation Method
4. D) Lower Right Area Method
2. Which endpoint of the subinterval is used in RRAM to determine the height of the rectangle?
1. A) Left endpoint
2. B) Right endpoint
3. C) Midpoint
4. D) Average of left and right endpoints
3. What is the formula for calculating $\Delta x$ in Riemann Sums, given the interval $[a, b]$ and $n$ subintervals?
1. A) $\Delta x = \frac{n}{b-a}$
2. B) $\Delta x = \frac{b-a}{n}$
3. C) $\Delta x = (b-a)n$
4. D) $\Delta x = \frac{a-b}{n}$
4. In MRAM, which point within each subinterval is used to determine the height of the rectangle?
1. A) Left endpoint
2. B) Right endpoint
3. C) Midpoint
4. D) Any random point
5. If you increase the number of subintervals ($n$) in a Riemann Sum, what generally happens to the accuracy of the approximation?
1. A) Accuracy decreases
2. B) Accuracy increases
3. C) Accuracy stays the same
4. D) Accuracy oscillates randomly
6. Using LRAM with 4 subintervals, approximate the area under the curve $f(x) = x^2$ from $x = 0$ to $x = 2$.
1. A) 1.75
2. B) 2.25
3. C) 2.75
4. D) 3.25
7. Which Riemann Sum method generally provides a more accurate approximation of the area under a curve?
1. A) LRAM
2. B) RRAM
3. C) MRAM
4. D) They are all equally accurate