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Average Value and Volume

  1. Find the average value of \(y=\sec^2(x)\) on the interval \([0, \pi/4]\).

  2. Find the average value of y on the given interval \([0, \frac{5}{2}]\). (Hint: Don't need to simplify answer)

    \[ y = \begin{cases} \frac{x}{4} & \text{if } 0 < x \leq 2 \\ \frac{1}{x^2 - 7x + 12} & \text{if } 2 < x < \frac{5}{2} \end{cases} \]

  3. What is a value of c that is guaranteed by the mean value theorem for integrals for the function \(y=\frac{1}{\sqrt{x}}\) on the interval \([1, 4]\)?

  4. Find the volume of the region bounded by \(y = e^x\), \(x = 0\), and \(y = 3\) rotated about the x-axis.