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

---

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.

---

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] ]

---

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.

---

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.

---

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.

---

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.

Would you like detailed solutions for any of these? 😊