Suppose that A(x) =B(x)Rα(x)f(u)du. Use sweeping to find ddxB(x)Rα(x)f(u)du. You may use x as the time andα(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.

Given, A(x)=∫α(x)B(x)f(u)du ...(1) Now differentiate (1) with respect to x, we obtain ddx(A(x))=ddx∫α(x)B(x)f(u)duBy using Leibniz integral rule: ddx∫a(x)b(x)f(t)dt=f(b(x))ddx(b(x))-f(a(x))ddx(a(x))∴ A'(x)=ddx∫α(x)B(x)f(u)du =f(B(x))ddx(B(x))-f(α(x))ddx(α(x)) =f(B(x))(B'(x))-f(α(x))(α'(x))⇒ A'(x)=B'(x)f(B(x))-α'(x)f(α(x))Hence, ddx∫α(x)B(x)f(u)du=B'(x)f(B(x))-α'(x)f(α(x))