Calculus Problems

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  1. Find the area between the curves \(y = x\) and \(y = x^2\)

  2. Compute the area between \(y = \sin{x}\) and \(y = \cos{x}\) and the interval \([\frac{\pi}{4}, \frac{5\pi}{4}]\)

  3. Find the average value on \([0, 16]\) of \(f(x) = \sqrt{x}\)

  4. What is the average value of the function \(f(x) = 3x^2 - 2x\) on \([1, 4]\)

  5. Determine the volume of the solid generated by rotating the function about the x-axis on \([0,3]\)
    \(y = \sqrt{9 - x^2}\)

  6. Find the average value of the function \(h(x) = \cos^4{(x)}\sin(x)\) on \([0, \pi]\)

  7. Determine the volume of the solid generated by rotating the function about the y-axis on \([0,4]\)
    \(y = \sqrt{x}\)

  8. Use the shell method to determine the volume of the solid formed by rotating the region about the y axis.
    \(y = x^2 + 2\)
    \(y = 0\), \(x = 0\), \(x = 2\)

  9. Use the shell method to determine the volume formed by the bounded region rotated about the x-axis.
    \(y = x^2\), \(y = 0\), \(x = 2\)

  10. Use the disk method to find the volume of the solid of rotation by rotating the bounded area around the y-axis
    \(y = 2x^2\), \(y = 0\), \(x = 2\)

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