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Practice Problems for 4/24

  1. Write the following limit as a definite integral
    \[ \lim_{||P||\rightarrow 0} \sum_{k=1}^n \sqrt{c_k}, \] where \(P\) is a partition of the interval [5,10].

  2. In class we found the limit of the Riemann sums of \(f(x)=x^2\) on [0,1] with \(n\) equal subintervals using both left and right endpoints for the \(c_k\) is equal to 1/3. From this information, and from what you know about the graph of \(x^2\), determine the value of the following definite integrals.
    1. \( \int_0^1 x^2\, dx\)
    2. \( \int_{-1}^0 x^2\, dx\)
    3. \( \int_{-1}^1 x^2\, dx\)

  3. Find the value of \(\int_{-2}^3 |x|\, dx\) using the geometric interpretation of definite integrals, and any relevant properties of definite integrals.

  4. Suppose \(f\) is a function which is continuous for all \(x\). Given \(\int_0^{20}f(x)\, dx=15\) and \(\int_0^{12} f(x)\,dx=7\) find the value of \(\int_{12}^{20}\frac{1}{2}f(x)\, dx\).