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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. \( \sqrt[3]{x}\) is increasing on [1,8], so \(1 \leq \sqrt[3]{x} \leq \sqrt[3]{8}=2\). From the max-min property of integrals, this implies \[ 1(8-1)\leq \int_1^8 \sqrt[3]{x}\,dx \leq 2(8-1). \] That is \(\int_1^8 \sqrt[3]{x}\, dx \in [7,14]\).
  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. First, using the additive property for integrals we write \( \int_0^7g(x)\, dx=\int_0^5g(x)\, dx+\int_5^7g(x)\, dx\). Next, use the bounds on \(g\) along with the domination property of integrals to get \[ \int_0^5g(x)\, dx \leq \int_0^5 h(x)\, dx=4 \quad \textrm{and} \quad \int_5^7g(x)\, dx \leq \int_5^7f(x)\, dx=12. \] Although we do not know the exact value of \(\int_0^7g(x)\, dx\), from this we can conclude \[ \int_0^7g(x)\, dx \leq 4+12=16. \]
  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 \([-\pi,2\pi]\).
    4. \[ \int_{-\pi}^{2\pi} \sin(x)+x\, dx \]