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Question

Let f be the function defined by f(x)=4/(x^2).

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Approximate the value of 12(upper limit)∫1(lower limit) f(x) dx using a left Riemann sum with the subintervals [1,4], [4,8], and [8,12].

Find the value of 12∫1 f(x) dx.

Find the value of ∞∫1 f(x) dx or explain why it does not exist.

Find the value of 12∫1 [f(x)⋅lnx] dx. Show the work.

Answer 1

a) Δx = L₃ = 4/1²·(4 - 1) + 4/4²(8 - 4) + 4/8²(12 - 8) = 12 + 1 + 1/4 = 13.25

b) ∫₁¹²(4/x²)dx = - 4/x₁¹² = - 4/12 + 4/1 = 11/3 = 3.67

c) ∫₁^∞44/x²dx = lim ∫₁^a4/x²dx = lim (-4/x)₁^a = lim (4 - 4/a) = 4

a→∞ a→∞ a→∞

d) ∫₁¹²4lnx/x²dx = 4lnx(-1/x)₁¹² + ∫₁¹²4/x²dx = - (ln12)/3 - 4/x₁¹² = [11 - ln(12)]/3 = 2.8384