1. Bees are constantly moving in and out of two bee hives, A and B. The rate of change in the number of bees per day for the two hives are Q' (t) and Q' (t), respectively. Explain what the following expression represents. 5 [Q'₁(t) - QB (t) dt Is it the same as the area between the Q' (t) and the Q' (t) curves?
1. Bees are constantly moving in and out of two bee hives, A and B. The rate of change in the number of bees per day for the two hives are Q' (t) and Q' (t), respectively. Explain what the following expression represents. 5 [Q'₁(t) - QB (t) dt Is it the same as the area between the Q' (t) and the Q' (t) curves?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Problem 1: Bee Movement Between Hives
**Context:**
Bees are constantly moving in and out of two bee hives, A and B. The rate of change in the number of bees per day for the two hives are \(Q'_A(t)\) and \(Q'_B(t)\), respectively.
**Task:**
Explain what the following expression represents:
\[ \int_{0}^{5} \left(Q'_A(t) - Q'_B(t)\right) dt \]
Is it the same as the area between the \(Q'_A(t)\) and \(Q'_B(t)\) curves?
**Explanation:**
The integral given, \(\int_{0}^{5} \left(Q'_A(t) - Q'_B(t)\right) dt\), represents the net change in the number of bees between the two hives over the time interval from \(t = 0\) to \(t = 5\) days.
**Step-by-Step Analysis:**
1. **Understanding \(Q'_A(t)\) and \(Q'_B(t)\):**
- \(Q'_A(t)\): The rate of change in the number of bees for Hive A at time \(t\).
- \(Q'_B(t)\): The rate of change in the number of bees for Hive B at time \(t\).
2. **Net Rate of Change:**
- \(Q'_A(t) - Q'_B(t)\) gives the difference in the rates of change of the number of bees between Hive A and Hive B at any time \(t\).
3. **Integral from 0 to 5:**
- The integral \(\int_{0}^{5} \left(Q'_A(t) - Q'_B(t)\right) dt\) accumulates these differences over the interval from \(t = 0\) to \(t = 5\).
4. **Interpretation:**
- This integral calculates the total net change in the number of bees between Hive A and Hive B over the 5-day period.
**Comparison with Area Between Curves:**
The expression is related to the area between the curves of \(Q'_A(t)\) and \(Q'_B(t)\). Specifically, the value of the integral represents the signed area between the curves of \(Q'_A(t)\) and \(Q'_B(t)\) over the interval](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff548c102-08e9-4b6b-8c12-787edeafe577%2Fcb8857a4-dbd8-4d32-984f-a8e9b0f61155%2Foatmw0o_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem 1: Bee Movement Between Hives
**Context:**
Bees are constantly moving in and out of two bee hives, A and B. The rate of change in the number of bees per day for the two hives are \(Q'_A(t)\) and \(Q'_B(t)\), respectively.
**Task:**
Explain what the following expression represents:
\[ \int_{0}^{5} \left(Q'_A(t) - Q'_B(t)\right) dt \]
Is it the same as the area between the \(Q'_A(t)\) and \(Q'_B(t)\) curves?
**Explanation:**
The integral given, \(\int_{0}^{5} \left(Q'_A(t) - Q'_B(t)\right) dt\), represents the net change in the number of bees between the two hives over the time interval from \(t = 0\) to \(t = 5\) days.
**Step-by-Step Analysis:**
1. **Understanding \(Q'_A(t)\) and \(Q'_B(t)\):**
- \(Q'_A(t)\): The rate of change in the number of bees for Hive A at time \(t\).
- \(Q'_B(t)\): The rate of change in the number of bees for Hive B at time \(t\).
2. **Net Rate of Change:**
- \(Q'_A(t) - Q'_B(t)\) gives the difference in the rates of change of the number of bees between Hive A and Hive B at any time \(t\).
3. **Integral from 0 to 5:**
- The integral \(\int_{0}^{5} \left(Q'_A(t) - Q'_B(t)\right) dt\) accumulates these differences over the interval from \(t = 0\) to \(t = 5\).
4. **Interpretation:**
- This integral calculates the total net change in the number of bees between Hive A and Hive B over the 5-day period.
**Comparison with Area Between Curves:**
The expression is related to the area between the curves of \(Q'_A(t)\) and \(Q'_B(t)\). Specifically, the value of the integral represents the signed area between the curves of \(Q'_A(t)\) and \(Q'_B(t)\) over the interval
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