Express the integral as a limit of Riemann sums using right endpoints. Do not evaluate the limit.

8

x2 +
(1/x)

dx

5

lim
n→∞
n

i=1

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!

Express the integral as a limit of Riemann sums using right endpoints. Do not evaluate the limit.

∫₅⁸ (x² + 1/x) dx

Learn how to express the integral ∫₅⁸ (x² + 1/x) dx as a limit of Riemann sums using right endpoints. This guide breaks down the process step-by-step, covering key calculus concepts such as definite integrals, Riemann sums, and limits. Perfect for high school and early college calculus students.

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