To express the integral

$\int_{5}^{8} \left( x^2 + \frac{1}{x} \right) \, dx$

as a limit of Riemann sums using right endpoints, follow these steps:

Step 1: Define the partition and subintervals

• The interval of integration is $[5, 8]$.
• Divide this interval into $n$ subintervals of equal width: $\Delta x = \frac{8 - 5}{n} = \frac{3}{n}.$
• Let $x_i = 5 + i \cdot \Delta x$ represent the right endpoints of each subinterval.

Step 2: Write the Riemann sum

The Riemann sum using right endpoints is: $S_n = \sum_{i=1}^{n} f(x_i) \Delta x,$ where $f(x) = x^2 + \frac{1}{x}$ and $\Delta x = \frac{3}{n}$.

Substituting $f(x_i)$ and $\Delta x$, the sum becomes: $S_n = \sum_{i=1}^{n} \left( (5 + i \cdot \Delta x)^2 + \frac{1}{5 + i \cdot \Delta x} \right) \Delta x.$

Step 3: Write the integral as a limit

Taking the limit as $n \to \infty$, the integral becomes: $\int_{5}^{8} \left( x^2 + \frac{1}{x} \right) \, dx = \lim_{n \to \infty} \sum_{i=1}^{n} \left( \left( 5 + \frac{3i}{n} \right)^2 + \frac{1}{5 + \frac{3i}{n}} \right) \cdot \frac{3}{n}.$

Final Answer:

$\int_{5}^{8} \left( x^2 + \frac{1}{x} \right) \, dx = \lim_{n \to \infty} \sum_{i=1}^{n} \left( \left( 5 + \frac{3i}{n} \right)^2 + \frac{1}{5 + \frac{3i}{n}} \right) \frac{3}{n}.$

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Do you want further details or have any questions?

Related Questions:

1. How would the expression change if left endpoints were used?
2. What are the practical applications of Riemann sums in real-world problems?
3. How do you determine the midpoint rule for a Riemann sum in this context?
4. Can the integral $\int_{5}^{8} \left( x^2 + \frac{1}{x} \right) dx$ be computed exactly using antiderivatives?
5. What is the significance of the $\Delta x$ term in a Riemann sum?

Tip: When practicing Riemann sums, always carefully define the function and subintervals to avoid calculation errors!