The table shows the rate of change of volume V with respect to time t (in liters per minute) of a balloon for selected times t, in minutes. 4 t dV dt i=1 ΣV'(ui) Ati 1 = 7 2.5 6 3 6 Find a left Riemann sum for on the interval [1, 5]. Use four subintervals and the values the table. dv dt (Use decimal notation. Give your answer to one decimal place.) 4 5 4
The table shows the rate of change of volume V with respect to time t (in liters per minute) of a balloon for selected times t, in minutes. 4 t dV dt i=1 ΣV'(ui) Ati 1 = 7 2.5 6 3 6 Find a left Riemann sum for on the interval [1, 5]. Use four subintervals and the values the table. dv dt (Use decimal notation. Give your answer to one decimal place.) 4 5 4
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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![### Calculus: Riemann Sums
#### Problem Statement
The table below shows the rate of change of volume \( V \) with respect to time \( t \) (in liters per minute) of a balloon for selected times \( t \), in minutes.
| \( t \) | 1 | 2.5 | 3 | 4 | 5 |
|----------------|-----|-----|----|----|----|
| \( \frac{dV}{dt} \) | 7 | 6 | 6 | 5 | 4 |
Find a left Riemann sum for \( \frac{dV}{dt} \) on the interval \([1, 5]\). Use four subintervals and the values in the table.
(Use decimal notation. Give your answer to one decimal place.)
### Solution
To find the left Riemann sum, we use the left endpoints for each subinterval. The subintervals from \([1, 5]\) are \([1, 2.5]\), \([2.5, 3]\), \([3, 4]\), and \([4, 5]\).
1. **Subinterval 1: \([1, 2.5]\)**
- Width \(\Delta t = 2.5 - 1 = 1.5\)
- Height \( \frac{dV}{dt} = 7 \) (at \( t = 1 \))
2. **Subinterval 2: \([2.5, 3]\)**
- Width \(\Delta t = 3 - 2.5 = 0.5\)
- Height \( \frac{dV}{dt} = 6 \) (at \( t = 2.5 \))
3. **Subinterval 3: \([3, 4]\)**
- Width \(\Delta t = 4 - 3 = 1\)
- Height \( \frac{dV}{dt} = 6 \) (at \( t = 3 \))
4. **Subinterval 4: \([4, 5]\)**
- Width \(\Delta t = 5 - 4 = 1\)
- Height \( \frac{dV](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe636121f-6982-47b6-a42b-1ffa9dd55a08%2F5065cfcb-f9cc-4f61-9694-e9a8aba0b94e%2Fgo3flkl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Calculus: Riemann Sums
#### Problem Statement
The table below shows the rate of change of volume \( V \) with respect to time \( t \) (in liters per minute) of a balloon for selected times \( t \), in minutes.
| \( t \) | 1 | 2.5 | 3 | 4 | 5 |
|----------------|-----|-----|----|----|----|
| \( \frac{dV}{dt} \) | 7 | 6 | 6 | 5 | 4 |
Find a left Riemann sum for \( \frac{dV}{dt} \) on the interval \([1, 5]\). Use four subintervals and the values in the table.
(Use decimal notation. Give your answer to one decimal place.)
### Solution
To find the left Riemann sum, we use the left endpoints for each subinterval. The subintervals from \([1, 5]\) are \([1, 2.5]\), \([2.5, 3]\), \([3, 4]\), and \([4, 5]\).
1. **Subinterval 1: \([1, 2.5]\)**
- Width \(\Delta t = 2.5 - 1 = 1.5\)
- Height \( \frac{dV}{dt} = 7 \) (at \( t = 1 \))
2. **Subinterval 2: \([2.5, 3]\)**
- Width \(\Delta t = 3 - 2.5 = 0.5\)
- Height \( \frac{dV}{dt} = 6 \) (at \( t = 2.5 \))
3. **Subinterval 3: \([3, 4]\)**
- Width \(\Delta t = 4 - 3 = 1\)
- Height \( \frac{dV}{dt} = 6 \) (at \( t = 3 \))
4. **Subinterval 4: \([4, 5]\)**
- Width \(\Delta t = 5 - 4 = 1\)
- Height \( \frac{dV
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