To approximate the area under the curve of the function $f(x) = \frac{x^3}{118} + \frac{x}{17} + \frac{112}{1}$ on the interval $[1, 5]$ using a Riemann sum with $n = 4$ rectangles, we need to calculate the sums using both the left and right endpoints.

1. Interval and width of each rectangle:

Given the interval $[1, 5]$, the width of each rectangle is:

$\Delta x = \frac{5 - 1}{4} = 1$

2. Left endpoint Riemann sum:

For the left endpoint, we evaluate the function at the left endpoint of each subinterval. The left endpoints of the subintervals for $n = 4$ are:

• $x_0 = 1$
• $x_1 = 2$
• $x_2 = 3$
• $x_3 = 4$

The left Riemann sum is given by:

$L_n = \sum_{i=0}^{n-1} f(x_i) \Delta x$

Now, we need to compute $f(x)$ at each of these left endpoints.

• $f(1) = \frac{1^3}{118} + \frac{1}{17} + \frac{112}{1} = \frac{1}{118} + \frac{1}{17} + 112$
• $f(2) = \frac{2^3}{118} + \frac{2}{17} + 112 = \frac{8}{118} + \frac{2}{17} + 112$
• $f(3) = \frac{3^3}{118} + \frac{3}{17} + 112 = \frac{27}{118} + \frac{3}{17} + 112$
• $f(4) = \frac{4^3}{118} + \frac{4}{17} + 112 = \frac{64}{118} + \frac{4}{17} + 112$

3. Right endpoint Riemann sum:

For the right endpoint, we evaluate the function at the right endpoint of each subinterval. The right endpoints of the subintervals for $n = 4$ are:

• $x_1 = 2$
• $x_2 = 3$
• $x_3 = 4$
• $x_4 = 5$

The right Riemann sum is given by:

$R_n = \sum_{i=1}^{n} f(x_i) \Delta x$

We need to compute $f(x)$ at each of these right endpoints:

• $f(2)$ has already been calculated.
• $f(3)$ has already been calculated.
• $f(4)$ has already been calculated.
• $f(5) = \frac{5^3}{118} + \frac{5}{17} + 112 = \frac{125}{118} + \frac{5}{17} + 112$

Now let me calculate these values and find the Riemann sums for both the left and right endpoints. Define the function f(x) = (x^3 / 118) + (x / 17) + 112 def f(x): return (x**3 / 118) + (x / 17) + 112

Left endpoint sum (for x = 1, 2, 3, 4)

left_endpoints = [1, 2, 3, 4] left_sum = sum(f(x) for x in left_endpoints)

Right endpoint sum (for x = 2, 3, 4, 5)

right_endpoints = [2, 3, 4, 5] right_sum = sum(f(x) for x in right_endpoints)

Calculate the sums by multiplying by delta x = 1 (since delta x = (5-1)/4)

delta_x = 1 left_riemann_sum = left_sum * delta_x right_riemann_sum = right_sum * delta_x

Round the results to two decimal places

left_riemann_sum_rounded = round(left_riemann_sum, 2) right_riemann_sum_rounded = round(right_riemann_sum, 2)

left_riemann_sum_rounded, right_riemann_sum_roundedThe approximate area under the curve using the Riemann sum with $n = 4$ rectangles is:

• Left endpoint Riemann sum: 449.44
• Right endpoint Riemann sum: 450.72

Would you like further details or have any questions about this process?

Here are a few related questions to deepen the understanding:

1. How does increasing the number of rectangles (increasing $n$) affect the accuracy of the Riemann sum?
2. What is the difference between a left endpoint and right endpoint Riemann sum in terms of approximation?
3. How would you calculate a Riemann sum using the midpoint of each interval?
4. How does the choice of function affect the accuracy of the Riemann sum approximation?
5. How do Riemann sums compare to using integration to find the exact area under a curve?

Tip: If you increase the number of rectangles $n$, the Riemann sum will become a better approximation of the actual integral.