Suppose that A(x) = B(x) R
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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Question
Suppose that A(x) =
B(x)
R
α(x)
f(u)du. Use sweeping to find d
dx
B(x)
R
α(x)
f(u)du. You may use x as the time and
α(x) and B(x) as the respective position function of the moving left and right vertical segments.
![**Transcription for Educational Website:**
---
**5. Understanding the Function \( A(x) \)**
Suppose that
\[
A(x) = \int_{\alpha(x)}^{B(x)} f(u) \, du.
\]
Utilize the concept of sweeping to determine
\[
\frac{d}{dx} \int_{\alpha(x)}^{B(x)} f(u) \, du.
\]
In this context, you may use \( x \) as the time variable, with \( \alpha(x) \) and \( B(x) \) representing the respective position functions of the moving left and right vertical segments.
**Graphical Explanation:**
The accompanying figure illustrates the application of the integral:
- The curve labeled \( f \) represents the function being integrated.
- The vertical lines at \( \alpha(x) \) and \( B(x) \) signify the dynamic limits of the integral, effectively ‘sweeping’ across the area under the curve as \( x \) changes.
- As \( \alpha(x) \) and \( B(x) \) shift, the area under the curve \( f \) between these points changes, corresponding to variations in \( A(x) \).
This visual demonstrates how changes in \( x \) affect the computed integral by altering the bounds from \( \alpha(x) \) to \( B(x) \).
---
This exploration will aid in grasping the calculus concept of integrating functions with variable limits and the resulting impact on the derivative of such an integral.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F61166d61-38cb-42ed-a6be-02009a56cf7f%2F7218333f-e245-4b43-9919-2be8754bb6bd%2Fw33imn2_processed.png&w=3840&q=75)
Transcribed Image Text:**Transcription for Educational Website:**
---
**5. Understanding the Function \( A(x) \)**
Suppose that
\[
A(x) = \int_{\alpha(x)}^{B(x)} f(u) \, du.
\]
Utilize the concept of sweeping to determine
\[
\frac{d}{dx} \int_{\alpha(x)}^{B(x)} f(u) \, du.
\]
In this context, you may use \( x \) as the time variable, with \( \alpha(x) \) and \( B(x) \) representing the respective position functions of the moving left and right vertical segments.
**Graphical Explanation:**
The accompanying figure illustrates the application of the integral:
- The curve labeled \( f \) represents the function being integrated.
- The vertical lines at \( \alpha(x) \) and \( B(x) \) signify the dynamic limits of the integral, effectively ‘sweeping’ across the area under the curve as \( x \) changes.
- As \( \alpha(x) \) and \( B(x) \) shift, the area under the curve \( f \) between these points changes, corresponding to variations in \( A(x) \).
This visual demonstrates how changes in \( x \) affect the computed integral by altering the bounds from \( \alpha(x) \) to \( B(x) \).
---
This exploration will aid in grasping the calculus concept of integrating functions with variable limits and the resulting impact on the derivative of such an integral.
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