Generate practice questions for my usage of the Riemann sum formula

Here are five practice questions to help you practice using the **Riemann sum formula**:

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### **1. Basic Left Riemann Sum**  
Approximate the area under the function \( f(x) = x^2 \) over the interval \( [1, 5] \) using a **left Riemann sum** with \( n = 4 \) subintervals.

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### **2. Right Riemann Sum with Unequal Intervals**  
Consider the function \( f(x) = \sin(x) \) on the interval \( [0, \frac{\pi}{2}] \). Use a **right Riemann sum** to estimate the integral with **three subintervals** of unequal widths:  
\[
\left[0, \frac{\pi}{6}\right], \quad \left[\frac{\pi}{6}, \frac{\pi}{4}\right], \quad \left[\frac{\pi}{4}, \frac{\pi}{2}\right]
\]

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### **3. Midpoint Riemann Sum for a Trigonometric Function**  
Approximate the integral  
\[
\int_0^2 (1 + \cos(\pi x)) \, dx
\]  
using a **midpoint Riemann sum** with \( n = 4 \) subintervals.

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### **4. Increasing the Number of Partitions**  
For \( f(x) = e^x \) on \( [0,1] \), compute the **left Riemann sum** with **\( n = 2, 4, 8 \)** and compare the approximations to the exact integral  
\[
\int_0^1 e^x \, dx.
\]  
Observe what happens as \( n \) increases.

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### **5. General Riemann Sum Formula Application**  
The function \( f(x) = x^3 - 4x \) is given over the interval \( [-2,2] \). Write the **general Riemann sum formula** for \( n \) subintervals and express it in summation notation.

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#### **Tip:**  
When using Riemann sums, always **check if the function is increasing or decreasing** over the interval. This helps determine whether the left or right sum will **overestimate or underestimate** the actual integral.

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Explore practice problems using the Riemann sum formula for approximating integrals. These questions cover various types of Riemann sums, including left, right, and midpoint, along with applications of the general Riemann sum formula. Ideal for high school students in Grades 11-12 learning integration techniques.

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