Chapter 5.9

Notes

Linguistic Mapping Exercises

Knowledge Checks

  1. A partition \(P\) of the set \([-2,5]\) has domain \(\{0,1,2,3\}\) and \(P=(-2,-1,2,5)\). Determine the mesh of \(P\).
  2. Identify a midpoint tagging \(\tau\) of this partition \(P\) of \([-2,5]\) that has three intervals and \(P=(-2,-1,2,5).\)
  3. Calculate the area of each region \(R\) formed by bounding the following.
    1. \(y=6\), \(x=2\), \(x=10\) and the \(x\)-axis
    2. \(y=x-2\), \(x=0\), \(x=5\) and the \(x\)-axis
    3. \(y=\sqrt{16-x^2}\), \(x=0\), \(x=4\) and the \(x\)-axis
    4. \(y=25\sqrt{16-x^2}\), \(x=0\), \(x=4\) and the \(x\)-axis.
  4. Take \(R\) to be the region formed by bounding \(y=4x^2+1\), \(x\)-axis, \(x=2\), and \(x=4.\) Approximate \(R\) by using six left-endpoint rectangles, six right-endpoint rectangles, and six midpoint rectangles.
  5. Take \(f\) to be the function given by \(f(x)=4x^2+1\) and the interval \(I\) to be the interval \(I=[-2,5].\) Determine an even partition \(P\) of \(I\) with five intervals and a left tagging \(\tau\) for \(P\) and calculate the quantity \(\mathcal{R}(f,P,\tau).\)
  6. Calculate using Riemann sums the area of the region \(R\) formed by bounding \(y=4x^2+1\), \(x=2\), \(x=4\), and the \(x\)-axis.
  7. Calculate the following by interpreting the definite integral as signed area.
    1. \(\displaystyle\int_{1}^{3}7\,\mathrm{d}x\)
    2. \(\displaystyle\int_{0}^{4}\sqrt{16-x^2}\,\mathrm{d}x\)
    3. \(\displaystyle\int_{2}^{4}(4x^2+1)\,\mathrm{d}x\)
  8. Take \(f\) and \(g\) to be integrable functions with \[\displaystyle\int^{5}_{2}f(x)\,\mathrm{d}x=-2\quad\text{and}\quad \int^{5}_{2}g(x)\,\mathrm{d}x=6.\] Calculate the following: \[\int_{2}^{5}(2f(x)+g(x)+4)\,\mathrm{d}x.\]
  9. Take \(f\) to be an integrable function with \[2\leq f(x)\leq 7\] on the interval \([-2,5].\) Determine an upper and lower bound for the following:
    1. \(\displaystyle\int_{0}^{3}f(x)\,\mathrm{d}x\)
    2. \(\displaystyle\int_{-2}^{5}f(x)\,\mathrm{d}x\)
    3. \(\displaystyle\int_{-2}^{5}2f(x)\,\mathrm{d}x\)
    4. \(\displaystyle\int_{-2}^{5}(f(x)+3)\,\mathrm{d}x\)
  10. Calculate the following:
    1. \(\displaystyle\int_{-2}^{3}x^4\,\mathrm{d}x\)
    2. \(\displaystyle\int_{-2}^{3}x^\frac{1}{3}\,\mathrm{d}x\)

Remember, do the knowledge checks before checking the solutions.

Practice Problems

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