Suppose that ƒ is continuous and nonnegative over [a, b],

Chapter 5, Problem 86E

(choose chapter or problem)

Suppose that \(f \\\) is continuous and nonnegative over \({[a, b]} \\\), as in the accompanying figure. By inserting points

    \(x_{1}, x_{2}, \ldots, x_{k-1}, x_{k}, \ldots, x_{n-1} \\\)

as shown, divide \({[a, b]} \\\) into \(n\) subintervals of lengths \(\Delta x_{1}=x_{1}-a \\\), \(\Delta x_{2}=x_{2}-x_{1}, \ldots, \Delta x_{n}=b-x_{n-1} \\\), which need not be equal.


a. If \(m_{k}=\min \{f(x)\) for \(x\) in the \(k\)th subinterval }, explain the connection between the lower sum

     \(L=m_{1} \Delta x_{1}+m_{2} \Delta x_{2}+\cdots+m_{n} \Delta x_{n} \\\)

and the shaded regions in the first part of the figure.

b. If \(M_{k}=\max \{f(x) \\\)\) for \(x\) in the \(k\)th subinterval} , explain the connection between the upper sum

     \(U=M_{1} \Delta x_{1}+M_{2} \Delta x_{2}+\cdots+M_{n} \Delta x_{n}\)

and the shaded regions in the second part of the figure.


c. Explain the connection between \(U-L\) and the shaded regions along the curve in the third part of the figure.

Equation Transcription:

 

Text Transcription:

f

[a,b]

n

x_1, x_2,..,x_k-1,x_k,..,x_n-1

delta x_1= x_1- a  

delta x_2 = x _2-x1,...delta x_n = b-x_n-1  

m_k = min{f(x)

x  

k

L=m_1 delta x_1 + m_2 delta x_2 +... + m_n delta x_n  

M_k=max{f(x)

U=M_1 delta x_1 + M_2 delta x_2 + ...+ M_n delta x_n

U-L

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