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Section Assignment #10 -Done Inclass 4/24

  1. Find both an upper and lower bound for the value of the integral \(\int_1^8 \sqrt[3]{x}\, dx\).

  2. Suppose you know \(g(x) \leq f(x)\) for \(x\in[5,7]\), \(g(x)\leq h(x)\) for \(x\in[0,5]\), \(\int_5^7 f(x)\, dx=12\) and \(\int_0^5 h(x)\, dx=4\).
    What can you say about the value of the integral \(\int_0^7 g(x)\, dx\)?

  3. Express the following limit as a definite integral \[\lim_{||P|| \rightarrow 0} \sum_{k=1}^n (\sin(c_k)+c_k\; )\Delta x_k, \] where \(P\) is a partition of the interval \([-\pi,2\pi]\).