Determine a region whose area is equal to the given limit. Do not evaluate the limit. lim Σ Σ(1+2) i = 1 O y = (1 + x) on [1, 3] Oy (1 + x) on [0, 2] 12 Oy = (1 + x) ¹2 on [1, 3] O y = x8 on [0, 2] Oy = (1 + x)8 on [0, 2] ³
Determine a region whose area is equal to the given limit. Do not evaluate the limit. lim Σ Σ(1+2) i = 1 O y = (1 + x) on [1, 3] Oy (1 + x) on [0, 2] 12 Oy = (1 + x) ¹2 on [1, 3] O y = x8 on [0, 2] Oy = (1 + x)8 on [0, 2] ³
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Determine a Region Whose Area is Equal to the Given Limit**
Determine a region whose area is equal to the given limit. Do not evaluate the limit.
\[
\lim_{{n \to \infty}} \sum_{{i=1}}^n \frac{2}{n} \left( 1 + \frac{2i}{n} \right)^8
\]
Options:
1. \( y = (1 + x)^7 \) on \([1, 3]\)
2. \( y = (1 + x)^7 \) on \([0, 2]\)
3. \( y = (1 + x)^{12} \) on \([1, 3]\)
4. \( y = x^8 \) on \([0, 2]\)
5. \( y = (1 + x)^8 \) on \([0, 2]\)
---
To determine which region’s area is equal to the given limit, match the integral form to the limit form. The integral form of the area under a curve \( f(x) \) from \( a \) to \( b \) is given by:
\[
\int_a^b f(x) \, dx
\]
The given limit suggests using the form of the Riemann sum for an integral. This can help identify the corresponding function and interval:
\[
\int_a^b f(x) \, dx = \lim_{{n \to \infty}} \sum_{{i=1}}^n f(x_i^*) \Delta x
\]
Comparing the Riemann sum to the given limit:
* \(\Delta x = \frac{2}{n}\)
* \(f\left(x_i\right) = \left(1 + \frac{2i}{n}\right)^8\)
* The interval \([a, b]\) is \([0, 2]\)
Thus:
* \(y = (1 + x)^8\) matches the function form.
**Conclusion:**
The correct region whose area is equal to the given limit is represented by the function \(y = (1 + x)^8\) on the interval \([0, 2]\).
So, the correct option is:
\[
\circ \quad y = (1 + x)^8 \text{ on } [0,](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa80fd0a7-9ae8-4bc5-b926-d69bfc10c0d7%2Fde5c9d15-52f7-4660-a91c-419680e0b6d0%2F00wcozj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Determine a Region Whose Area is Equal to the Given Limit**
Determine a region whose area is equal to the given limit. Do not evaluate the limit.
\[
\lim_{{n \to \infty}} \sum_{{i=1}}^n \frac{2}{n} \left( 1 + \frac{2i}{n} \right)^8
\]
Options:
1. \( y = (1 + x)^7 \) on \([1, 3]\)
2. \( y = (1 + x)^7 \) on \([0, 2]\)
3. \( y = (1 + x)^{12} \) on \([1, 3]\)
4. \( y = x^8 \) on \([0, 2]\)
5. \( y = (1 + x)^8 \) on \([0, 2]\)
---
To determine which region’s area is equal to the given limit, match the integral form to the limit form. The integral form of the area under a curve \( f(x) \) from \( a \) to \( b \) is given by:
\[
\int_a^b f(x) \, dx
\]
The given limit suggests using the form of the Riemann sum for an integral. This can help identify the corresponding function and interval:
\[
\int_a^b f(x) \, dx = \lim_{{n \to \infty}} \sum_{{i=1}}^n f(x_i^*) \Delta x
\]
Comparing the Riemann sum to the given limit:
* \(\Delta x = \frac{2}{n}\)
* \(f\left(x_i\right) = \left(1 + \frac{2i}{n}\right)^8\)
* The interval \([a, b]\) is \([0, 2]\)
Thus:
* \(y = (1 + x)^8\) matches the function form.
**Conclusion:**
The correct region whose area is equal to the given limit is represented by the function \(y = (1 + x)^8\) on the interval \([0, 2]\).
So, the correct option is:
\[
\circ \quad y = (1 + x)^8 \text{ on } [0,
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