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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Question
![**Express the Limit as an Integral**
Given the following limit expression:
\[
\lim_{{n \to \infty}} \sum_{{i=1}}^{n} \left( 7(x_i^*)^2 - 2(x_i^*)^3 \right) \Delta x \text{ over } [0, 4]
\]
we are asked to express this limit as an integral.
**Correct Integral Form:**
The correct form of the integral is:
\[
\int_{0}^{4} \left( 7x^2 - 2x^3 \right) \, dx
\]
**Error Identification:**
- The image includes a box with \((7x^2 - x^3)\) next to a red 'x', indicating an error.
- The function inside the integral must be \(7x^2 - 2x^3\) instead of \(7x^2 - x^3\).
The integral correctly representing this limit should thus be:
\[
\int_{0}^{4} \left( 7x^2 - 2x^3 \right) \, dx
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F70a4635b-5071-4f8c-8108-030e9029d2c6%2F36f7d8d8-5cf0-4a55-8d6a-d5d3b9baa208%2Fs4q0su6_processed.png&w=3840&q=75)
Transcribed Image Text:**Express the Limit as an Integral**
Given the following limit expression:
\[
\lim_{{n \to \infty}} \sum_{{i=1}}^{n} \left( 7(x_i^*)^2 - 2(x_i^*)^3 \right) \Delta x \text{ over } [0, 4]
\]
we are asked to express this limit as an integral.
**Correct Integral Form:**
The correct form of the integral is:
\[
\int_{0}^{4} \left( 7x^2 - 2x^3 \right) \, dx
\]
**Error Identification:**
- The image includes a box with \((7x^2 - x^3)\) next to a red 'x', indicating an error.
- The function inside the integral must be \(7x^2 - 2x^3\) instead of \(7x^2 - x^3\).
The integral correctly representing this limit should thus be:
\[
\int_{0}^{4} \left( 7x^2 - 2x^3 \right) \, dx
\]
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