What is the formula for the total distance traveled by an object with a velocity v(t) = sin(t) on the interval [0, 4]? 4 sin (t) dt – - ¹ sin (t) dt + sin (t) dt ₁ sin (t) dt 4 So sin (t) dt - f sin (t) dt - - f sin (t) dt + f sin (t) dt ㅠ
What is the formula for the total distance traveled by an object with a velocity v(t) = sin(t) on the interval [0, 4]? 4 sin (t) dt – - ¹ sin (t) dt + sin (t) dt ₁ sin (t) dt 4 So sin (t) dt - f sin (t) dt - - f sin (t) dt + f sin (t) dt ㅠ
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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![### Problem Statement:
What is the formula for the total distance traveled by an object with a velocity \( v(t) = \sin(t) \) on the interval \([0, 4]\)?
### Options:
1. \(\int_{0}^{\pi} \sin(t) \, dt - \int_{\pi}^{4} \sin(t) \, dt\)
2. \(-\int_{0}^{1} \sin(t) \, dt + \int_{1}^{4} \sin(t) \, dt\)
3. \(\int_{1}^{0} \sin(t) \, dt - \int_{4}^{1} \sin(t) \, dt\)
4. \(-\int_{0}^{\pi} \sin(t) \, dt + \int_{4}^{\pi} \sin(t) \, dt\)
### Explanation:
#### Correct Answer:
The formula given in the highlighted answer is:
\[ \int_{0}^{\pi} \sin(t) \, dt - \int_{\pi}^{4} \sin(t) \, dt \]
#### Details:
On the interval [0, π], the sine function \( \sin(t) \) is non-negative, contributing positively to the distance traveled. On the interval \([π, 4]\), the sine function \( \sin(t) \) is non-positive, contributing negatively to the total displacement. Thus, the total distance traveled considering the absolute value is computed as follows:
1. **First Interval (0 to π):**
\(\int_{0}^{\pi} \sin(t) \, dt\)
2. **Second Interval (π to 4):**
\(-\int_{\pi}^{4} \sin(t) \, dt\)
This total formula accounts for the change in sign of \(\sin(t)\) over the intervals, ensuring that the total distance is properly computed as the sum of absolute changes rather than mere displacement.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb111f9a2-d607-44d0-9cf1-983102308b09%2Fcd82321e-0541-452c-8350-55acfb107280%2F3edsq_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement:
What is the formula for the total distance traveled by an object with a velocity \( v(t) = \sin(t) \) on the interval \([0, 4]\)?
### Options:
1. \(\int_{0}^{\pi} \sin(t) \, dt - \int_{\pi}^{4} \sin(t) \, dt\)
2. \(-\int_{0}^{1} \sin(t) \, dt + \int_{1}^{4} \sin(t) \, dt\)
3. \(\int_{1}^{0} \sin(t) \, dt - \int_{4}^{1} \sin(t) \, dt\)
4. \(-\int_{0}^{\pi} \sin(t) \, dt + \int_{4}^{\pi} \sin(t) \, dt\)
### Explanation:
#### Correct Answer:
The formula given in the highlighted answer is:
\[ \int_{0}^{\pi} \sin(t) \, dt - \int_{\pi}^{4} \sin(t) \, dt \]
#### Details:
On the interval [0, π], the sine function \( \sin(t) \) is non-negative, contributing positively to the distance traveled. On the interval \([π, 4]\), the sine function \( \sin(t) \) is non-positive, contributing negatively to the total displacement. Thus, the total distance traveled considering the absolute value is computed as follows:
1. **First Interval (0 to π):**
\(\int_{0}^{\pi} \sin(t) \, dt\)
2. **Second Interval (π to 4):**
\(-\int_{\pi}^{4} \sin(t) \, dt\)
This total formula accounts for the change in sign of \(\sin(t)\) over the intervals, ensuring that the total distance is properly computed as the sum of absolute changes rather than mere displacement.
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