To calculate the left and right Riemann sums for $f(x) = x + 2$ over the interval $[2, 7]$ with $n = 5$, we will proceed as follows:

1. Determine the width of each subinterval ($\Delta x$): $\Delta x = \frac{7 - 2}{5} = 1$

2. Left Riemann Sum: For the left Riemann sum, we use the function values at the left endpoints of each subinterval. The left endpoints are $x = 2, 3, 4, 5,$ and $6$.

   • $f(2) = 2 + 2 = 4$
   • $f(3) = 3 + 2 = 5$
   • $f(4) = 4 + 2 = 6$
   • $f(5) = 5 + 2 = 7$
   • $f(6) = 6 + 2 = 8$

   The left Riemann sum is then: $L_5 = \Delta x \cdot (f(2) + f(3) + f(4) + f(5) + f(6))$ $L_5 = 1 \cdot (4 + 5 + 6 + 7 + 8) = 1 \cdot 30 = 30$

3. Right Riemann Sum: For the right Riemann sum, we use the function values at the right endpoints of each subinterval. The right endpoints are $x = 3, 4, 5, 6,$ and $7$.

   • $f(3) = 3 + 2 = 5$
   • $f(4) = 4 + 2 = 6$
   • $f(5) = 5 + 2 = 7$
   • $f(6) = 6 + 2 = 8$
   • $f(7) = 7 + 2 = 9$

   The right Riemann sum is then: $R_5 = \Delta x \cdot (f(3) + f(4) + f(5) + f(6) + f(7))$ $R_5 = 1 \cdot (5 + 6 + 7 + 8 + 9) = 1 \cdot 35 = 35$

Summary:

• Left Riemann Sum ($L_5$): 30
• Right Riemann Sum ($R_5$): 35

Would you like more details on this process or have any other questions?

Here are some related questions you might explore:

1. What is the average of the left and right Riemann sums, and how does it relate to the midpoint Riemann sum?
2. How would the Riemann sums change if $n$ were increased, say to $n = 10$?
3. Can we calculate the exact area under $f(x) = x + 2$ from $x = 2$ to $x = 7$ using integration?
4. How would we set up a trapezoidal sum for this function on the interval $[2, 7]$ with $n = 5$?
5. How do left and right Riemann sums compare in accuracy for different types of functions?

Tip: Increasing $n$ generally improves the accuracy of Riemann sums by providing a closer approximation to the actual area.