Answer to Question #249443 in Calculus for Akiller

Question #249443

Show that π‘ˆ(βˆ’π‘“, 𝑝) = βˆ’πΏ(𝑓, 𝑝) and 𝐿(βˆ’π‘“, 𝑝) = βˆ’π‘ˆ(𝑓, 𝑝)


1
Expert's answer
2021-10-11T10:58:25-0400

π‘ˆ(𝑓, 𝑝) and 𝐿(𝑓, 𝑝) are upper and lower Riemann sums for the partition p.

"\ud835\udc48(\ud835\udc53, \ud835\udc5d)=\\sum M_i\\Delta x_i,\\ L(\ud835\udc53, \ud835\udc5d)=\\sum m_i\\Delta x_i"

"M_i=sup\\{f(x):x_{i-1}\\le x \\le x_i\\},\\ m_i=inf\\{f(x):x_{i-1}\\le x \\le x_i\\}"


"sup\\{-f(x):x_{i-1}\\le x \\le x_i\\}=-inf\\{f(x):x_{i-1}\\le x \\le x_i\\}=-m_i"

"U(-f,p)=-\\sum m_i\\Delta x_i=-L(f,p)"


"inf\\{-f(x):x_{i-1}\\le x \\le x_i\\}=-sup\\{f(x):x_{i-1}\\le x \\le x_i\\}=-M_i"

"L(-f,p)=-\\sum M_i\\Delta x_i=-U(f,p)"

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