Chapter 5.9

Notes

Linguistic Mapping Exercises

Knowledge Checks

  1. 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.
  2. Approximate the region \(R\) given below by using six left-endpoint rectangles, six right-endpoint rectangles, and six midpoint rectangles. The region \(R\) formed by bounding \(y=4x^2+1\), \(x\)-axis, \(x=2\), and \(x=4.\)
  3. Calculate the area of the region \(R\) formed by bounding \(y=4x^2+1\), \(x=2\) and \(x=4.\)
  4. 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. \(\displaystyle\int_{2}^{4}(4x^2+1)\,\mathrm{d}x\)
  5. Suppose that \(f\) and \(g\) are integrable and \[\displaystyle\int^{5}_{2}f(x)\,\mathrm{d}x=-2\quad\text{and}\quad \int^{5}_{2}g(x)\,\mathrm{d}x=6.\] Calculate the following: \[\int_{2}^{5}(2f(x)+g(x)+4)\,\mathrm{d}x.\]
  6. Suppose that \(f\) is integrable and \[2\leq f(x)\leq 7\] on the interval \([-2,5].\) Calculate 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. \(\displaystyle\int_{-2}^{5}(f(x)+3)\,\mathrm{d}x\)

Practice Problems

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