## Understanding the Symbols in a Riemann Sum and Their Connection to Definite Integrals### Mathematical RepresentationThe 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

“Since you have posted a question with multiple sub parts, we will provide the solution only to the…

Answered: (a) S (b) a (c) b (d) f(t) (e) dt (f) n…

(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.

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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