86. Suppose that ƒ is continuous and nonnegative over [a, b] , as in the accompanying figure.

Chapter 5.3  The Definite Integral Ex86
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86. Suppose that ƒ is continuous and nonnegative over [a, b] , as in the accompanying figure. By inserting points x1, x2,c, xk-1, xk , …, xn-1 as shown, divide [a, b] into n subintervals of lengths ∆x1 = x1 - a, ∆x2 = x2 - x1,c, ∆xn = b - xn-1, which need not be equal. a. If mk = min 5ƒ(x) for x in the kth subinterval6, explain the connection between the lower sum L = m1 ∆x1 + m2 ∆x2 + g+ mn ∆xn and the shaded regions in the first part of the figure. b. If Mk = max 5ƒ(x) for x in the kth subinterval6, explain the connection between the upper sum U = M1 ∆x1 + M2 ∆x2 + g+ Mn ∆xn 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.