Let 0 :s a < b, let Q(x) := x2 for x e [a, b] and let l' := {[Xj_1'xj]}7=1be a partition

Chapter 7, Problem 16

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Let 0 :s a < b, let Q(x) := x2 for x e [a, b] and let l' := {[Xj_1'xj]}7=1be a partition of [a, b].For each i, let qi be the positive square root ofI (2 23 xj + XjXj_1 + xj_1 ) .(a) Show that qi satisfies 0 :s Xi_I :s qi :s Xj'(b) Show that Q(qi)(Xi - xi_I) = t(x; - xLI)'(c) If Q is the tagged partition with the same subintervals as l' and the tags qi' show thatS(Q; Q) = t(b3 - a3).(d) Use the argument in Example 7.1.3(c) to show that Q e 'R[a, b] andlb Q = lb x2dx = t(b3 - a3).