(a) S (b) a (c) b (d) f(t) (e) dt (f) n (h) n-1 (i) f(t;) (j) Δt (k) lim 848 ned the standard 12.
(a) S (b) a (c) b (d) f(t) (e) dt (f) n (h) n-1 (i) f(t;) (j) Δt (k) lim 848 ned the standard 12.
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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![## Understanding the Symbols in a Riemann Sum and Their Connection to Definite Integrals
### Mathematical Representation
The definite integral of a function \( f(t) \) from \( a \) to \( b \) is given by:
\[
\int_a^b f(t) \, dt = \lim_{n \to \infty} \sum_{i=0}^{n-1} f(t_i) \Delta t
\]
### Explanation of Each Symbol
#### (a) \(\int\)
- **Purpose**: Represents the integral symbol, indicating the summation of infinite infinitesimal elements, \( f(t) \, dt \), over the interval \( [a, b] \).
#### (b) \(a\)
- **Purpose**: Lower limit of integration, representing the starting point of the interval on the real line.
#### (c) \(b\)
- **Purpose**: Upper limit of integration, representing the endpoint of the interval on the real line.
#### (d) \(f(t)\)
- **Purpose**: The function to be integrated, expressing the relationship between the variable \( t \) and the function's output.
#### (e) \(dt\)
- **Purpose**: Infinitesimal change in the variable \( t \), used to denote that integration is being performed with respect to \( t \).
#### (f) \(n\)
- **Purpose**: Number of subdivisions of the interval \([a, b]\), which approaches infinity in a Riemann sum.
#### (g) \(i\)
- **Purpose**: Index of summation, used to identify each subinterval and its corresponding evaluation point, \( t_i \).
#### (h) \(n-1\)
- **Purpose**: Shows that the sum includes all terms from 0 up to \( n-1 \), corresponding to the subdivisions of \([a, b]\).
#### (i) \(f(t_i)\)
- **Purpose**: Height of the rectangle at the subinterval associated with \( t_i \), where \( t_i \) is a sample point in the \( i \)-th subinterval.
#### (j) \(\Delta t\)
- **Purpose**: Width of each subinterval, calculated as \( \Delta t = \frac{b-a}{n} \).
#### (k) \(\lim_{n \to](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe1b1d143-3544-4565-ae01-c2b2100e15ce%2F06d875c3-e391-4b64-8f95-9efb238aab82%2F0o9zq48_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Understanding the Symbols in a Riemann Sum and Their Connection to Definite Integrals
### Mathematical Representation
The definite integral of a function \( f(t) \) from \( a \) to \( b \) is given by:
\[
\int_a^b f(t) \, dt = \lim_{n \to \infty} \sum_{i=0}^{n-1} f(t_i) \Delta t
\]
### Explanation of Each Symbol
#### (a) \(\int\)
- **Purpose**: Represents the integral symbol, indicating the summation of infinite infinitesimal elements, \( f(t) \, dt \), over the interval \( [a, b] \).
#### (b) \(a\)
- **Purpose**: Lower limit of integration, representing the starting point of the interval on the real line.
#### (c) \(b\)
- **Purpose**: Upper limit of integration, representing the endpoint of the interval on the real line.
#### (d) \(f(t)\)
- **Purpose**: The function to be integrated, expressing the relationship between the variable \( t \) and the function's output.
#### (e) \(dt\)
- **Purpose**: Infinitesimal change in the variable \( t \), used to denote that integration is being performed with respect to \( t \).
#### (f) \(n\)
- **Purpose**: Number of subdivisions of the interval \([a, b]\), which approaches infinity in a Riemann sum.
#### (g) \(i\)
- **Purpose**: Index of summation, used to identify each subinterval and its corresponding evaluation point, \( t_i \).
#### (h) \(n-1\)
- **Purpose**: Shows that the sum includes all terms from 0 up to \( n-1 \), corresponding to the subdivisions of \([a, b]\).
#### (i) \(f(t_i)\)
- **Purpose**: Height of the rectangle at the subinterval associated with \( t_i \), where \( t_i \) is a sample point in the \( i \)-th subinterval.
#### (j) \(\Delta t\)
- **Purpose**: Width of each subinterval, calculated as \( \Delta t = \frac{b-a}{n} \).
#### (k) \(\lim_{n \to
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