The following limit represents the definite integral ∫baf(x)dx for some real numbers a and b, and an integrable function f:

limn→∞∑i=1n4in2(2+4in)

**a)** Find Δx and xi from this expression and deduce the values of a and b, and the function f.

**FORMATTING:** *Your expression for Δx should be an expression involving n.  Your expression for xi  should be an expression involving the index i and n. The numbers a and b are real numbers, and f(x) should be a function of x.* *Write your answer in the form [Δx,xi,a,b,f(x)] including the square brackets [ ] and commas (,) between each term.*** *Strict** scientific calculator notation is required in your answer, meaning in particular *** for multiplication**, e.g. 2x must be written* **2*x**, *and (x+1)(x+2) must be written* **(x+1)*(x+2).**

Answer: [Δx,xi,a,b,f(x)]=

**b)** Compute the value of the limit above. Give your answer with an accuracy of four decimal places.

Answer: limn→∞∑i=1n4in2(2+4in)=

To solve this, we need to recognize that the sum resembles a Riemann sum for a definite integral of the form \( \int_a^b f(x) \, dx \).

### Part (a)
The general form of a Riemann sum for \( \int_a^b f(x) \, dx \) is:
\[
\lim_{n \to \infty} \sum_{i=1}^n f(x_i) \, \Delta x
\]
where:
- \( \Delta x = \frac{b - a}{n} \),
- \( x_i = a + i \, \Delta x \).

In our case, the expression inside the sum, \( \frac{4i}{n^2(2 + \frac{4i}{n})} \), suggests:
1. **\( \Delta x \)** is likely \( \frac{1}{n} \), as it is the term dividing \( f(x) \).
2. **\( x_i \)** is represented by \( \frac{4i}{n} \), because this term varies with \( i \) and scales with \( \frac{1}{n} \).

Given this, we deduce:
- \( \Delta x = \frac{1}{n} \),
- \( x_i = \frac{4i}{n} \).

Next, we identify \( a \), \( b \), and \( f(x) \):
- Since \( x_i = \frac{4i}{n} \) spans from \( \frac{4 \cdot 1}{n} \approx 0 \) to \( \frac{4 \cdot n}{n} = 4 \), we conclude \( a = 0 \) and \( b = 4 \).
- The function \( f(x) \) must match the form inside the sum, with \( x = \frac{4i}{n} \) substituted. Rewriting \( \frac{4i}{n^2(2 + \frac{4i}{n})} \) in terms of \( x = \frac{4i}{n} \), we get:
  \[
  f(x) = \frac{x}{2 + x}.
  \]

Thus, we have:
\[
\text{Answer to (a): } [\Delta x, x_i, a, b, f(x)] = \left[ \frac{1}{n}, \frac{4*i}{n}, 0, 4, \frac{x}{2 + x} \right].
\]

### Part (b)
Now, we compute the value of the limit, which corresponds to the definite integral:
\[
\int_0^4 \frac{x}{2 + x} \, dx.
\]

To evaluate this integral:
1. **Substitute** \( u = 2 + x \Rightarrow du = dx \).
2. When \( x = 0 \), \( u = 2 \); and when \( x = 4 \), \( u = 6 \).
3. This transforms the integral to:
   \[
   \int_2^6 \frac{u - 2}{u} \, du = \int_2^6 \left( 1 - \frac{2}{u} \right) du.
   \]
4. Now, integrate term-by-term:
   \[
   \int_2^6 1 \, du - \int_2^6 \frac{2}{u} \, du = \left[ u - 2 \ln |u| \right]_2^6.
   \]
5. Plugging in the limits:
   \[
   \left( 6 - 2 \ln 6 \right) - \left( 2 - 2 \ln 2 \right) = 4 - 2 \ln 3.
   \]
6. Calculating \( 4 - 2 \ln 3 \) to four decimal places:
   \[
   4 - 2 \ln 3 \approx 1.8028.
   \]

Thus, the answer for (b) is:
\[
\text{Answer: } \lim_{n \to \infty} \sum_{i=1}^n \frac{4i}{n^2(2 + \frac{4i}{n})} = 1.8028.
\]

Let me know if you need more details or have any questions. 

Here are some related questions to expand on this concept:
1. How can we identify the limits of integration in other Riemann sum problems?
2. What techniques can we use for more complicated integrals involving logarithms?
3. Why does the choice of \( x_i \) matter in Riemann sums?
4. What is the significance of \( \Delta x \) in terms of partitioning the interval?
5. How does substitution simplify complex integrals?

**Tip:** To recognize integrable functions in Riemann sums, match terms in the sum to an integral's format, then identify \( \Delta x \) and \( x_i \) accordingly.

The following limit represents the definite integral ∫baf(x)dx for some real numbers a and b, and an integrable function f: limn→∞∑i=1n4in2(2+4in) a) Find Δx and xi from this expression and deduce the values of a and b, and the function f. FORMATTING: Your expression for Δx should be an expression involving n. Your expression for xi should be an expression involving the index i and n. The numbers a and b are real numbers, and f(x) should be a function of x. Write your answer in the form [Δx,xi,a,b,f(x)] including the square brackets [ ] and commas (,) between each term. Strict scientific calculator notation is required in your answer, meaning in particular * for multiplication, e.g. 2x must be written 2*x, and (x+1)(x+2) must be written (x+1)*(x+2). Answer: [Δx,xi,a,b,f(x)]= b) Compute the value of the limit above. Give your answer with an accuracy of four decimal places. Answer: limn→∞∑i=1n4in2(2+4in)=

Explore how to deduce Δx, xi, and the limits of integration a and b for a given Riemann sum and evaluate the definite integral. This problem provides a step-by-step solution, involving key calculus concepts such as Riemann sums, limits, and substitution in integrals. Aimed at undergraduate calculus students.

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