The exercise in your screenshot involves approximating the area under the curve $f(x) = \sqrt{4 - x^2}$ over the interval $[-2, 2]$ using Riemann sums.

Here's a breakdown of the problem as it appears in the image:

1. Midpoint Riemann Sum with $n = 4$ subintervals:

   • The interval $[-2, 2]$ is divided into 4 equal parts, giving a subinterval width of $\Delta x = 1$.
   • Midpoints for the four subintervals are identified as:
     • $x_1^* = -1.5$
     • $x_2^* = -0.5$
     • $x_3^* = 0.5$
     • $x_4^* = 1.5$
   • These values are used to compute the Midpoint Riemann sum approximation, resulting in $6.52$ when rounded to two decimal places.

2. The next part of the exercise asks to approximate the area using a Left Riemann Sum with $n = 4$ subintervals. This approach would use the left endpoint of each interval instead of the midpoint.

Would you like a detailed explanation on calculating the Left Riemann Sum for this setup, or do you have other questions on Riemann sums?

Here are five related questions for further exploration:

1. How is the midpoint of each subinterval calculated in a Midpoint Riemann sum?
2. What would change if we used a Right Riemann sum instead?
3. How does the choice of Riemann sum (Left, Right, Midpoint) affect the approximation accuracy?
4. What is the exact area under the curve $f(x) = \sqrt{4 - x^2}$ over $[-2, 2]$ if calculated with integration?
5. How can we improve the approximation by increasing the number of subintervals $n$?

Tip: Increasing the number of subintervals in a Riemann sum generally improves the accuracy of the area approximation, as it provides a closer fit to the curve.