Give step-by-step solution with explanation and final answer:
I
f

I
=
∫
x
=
0
2
x
d
x
=
2
and the error value
i
s

E
(
n
)
=
R
S
(
n
)
-
1
,
what
i
s
the optimal error value?

Question

Give step-by-step solution with explanation and final answer: 
I
f
 
I
=
∫
x
=
0
2
x
d
x
=
2
 ﻿and the error value 
i
s
 
E
(
n
)
=
R
S
(
n
)
-
1
,
 ﻿what 
i
s
 ﻿the optimal error value?

VIPSolutions · Accepted Answer

Let's break down the problem step-by-step:

---

### 1. **Given Data**

- The integral:
  $$
  I = \int_{x=0}^{2} x \, dx = 2
  $$
- Error value defined as:
  $$
  E(n) = RS(n) - 1
  $$
  Where $ RS(n) $ is presumably the **right Riemann sum** approximation with $ n $ subdivisions.

---

### 2. **Evaluate the Integral**

$$
I = \int_{0}^{2} x \, dx = \left[ \frac{x^2}{2} \right]_0^2 = \frac{2^2}{2} - \frac{0^2}{2} = \frac{4}{2} = 2
$$
So the given value is confirmed.

---

### 3. **Right Riemann Sum for $ f(x) = x $ on [0, 2]**

Partition the interval $[0,2]$ into $ n $ equal subintervals:

- Width of each subinterval: $ \Delta x = \frac{2 - 0}{n} = \frac{2}{n} $
- Right endpoints: $ x_i = 0 + i \Delta x = \frac{2i}{n} $, $ i = 1, 2, ..., n $

The right Riemann sum is:
$$
RS(n) = \sum_{i=1}^{n} f(x_i) \Delta x = \sum_{i=1}^{n} x_i \Delta x = \sum_{i=1}^{n} \left(\frac{2i}{n}\right) \left(\frac{2}{n}\right) = \sum_{i=1}^{n} \frac{4i}{n^2}
$$
$$
RS(n) = \frac{4}{n^2} \sum_{i=1}^{n} i = \frac{4}{n^2} \cdot \frac{n(n+1)}{2} = \frac{4}{n^2} \cdot \frac{n(n+1)}{2}
$$
$$
= \frac{2n(n+1)}{n^2} = 2\left(1 + \frac{1}{n}\right)
$$

---

### 4. **The Error Value $ E(n) $**

Recall:
$$
E(n) = RS(n) - 1
$$
From above,
$$
RS(n) = 2 \left(1 + \frac{1}{n}\right) = 2 + \frac{2}{n}
$$
So,
$$
E(n) = 2 + \frac{2}{n} - 1 = 1 + \frac{2}{n}
$$

---

### 5. **Optimal Error Value**

**"Optimal error value"** typically means the minimum possible error, i.e., the value as $ n \rightarrow \infty $.

Take the limit as $ n \to \infty $:

$$
\lim_{n \to \infty} E(n) = \lim_{n \to \infty} \left(1 + \frac{2}{n}\right) = 1
$$

---

### **Summary Table**

| Step | Calculation                                              | Result                |
|------|---------------------------------------------------------|-----------------------|
| 1    | $ I = \int_0^2 x dx $                                 | 2                     |
| 2    | $ RS(n) = 2\left(1 + \frac{1}{n}\right) $             | $ 2 + \frac{2}{n} $ |
| 3    | $ E(n) = RS(n) - 1 $                                  | $ 1 + \frac{2}{n} $ |
| 4    | $ \lim_{n \to \infty} E(n) $ (optimal error value)    | **1**                 |

---

## **Final Answer**
$$
\boxed{1}
$$
The **optimal error value** is $ \boxed{1} $.

✨ VIPSolutions

Give step-by-step solution with explanation and final answer: I f I = ∫ x = 0 2 x d x = 2 and the error value i s E ( n ) = R S ( n ) - 1 , what i s the optimal error value?

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