BC Calculus Assignment 54: Area-Volume Problems

Directions: For each of the following, give the solution on a single, detachable sheet of paper (that is, the solution to number 1 is on one sheet, the solution to number 2 is on another sheet, etc.) Tomorrow in class, we'll switch solutions, I'll post the standards, and we'll "score" each other. (Just for practice. The "score" won't count.)

Questions came from Advanced Placement Calculus Tests as Indicated.
  1. (1998 BC 1 Calc active) Let \(R\) be the region in the first quadrant bounded by the graph of \(y = 8 - x^{\frac{3}{2}}\), the x-axis, and the y-axis.
    1. Find the area of the region \(R\).
    2. Find the volume of the solid generated when \(R\) is revolved about the x-axis.
    3. The vertical line \(x = k\) divides the region \(R\) into two regions such that when these two regions are revolved about the x-axis, they generate solids with equal volumes. Find the value of \(k\).
  2. (1999 BC 2 Calc active) Given the region \(R\) bounded by the graph of \(y = x^2\) and the line \(y = 4\),
    1. Find the area of \( R \).
    2. Find the volume of the solid generated by revolving \( R \) about the x-axis.
    3. There exists a number \(k, k > 4\), such that when \( R \) is revolved about the line \(y = k\), the resulting solid has the same volume as the solid in part (b). Write, but do not solve, an equation involving an integral expression that can be used to find the value of \(k\).
  3. (2000 BC 1 Calc active) Let \( R \) be the shaded region in the first quadrant enclosed by the graphs of \(y = e^{-x^2}, y = 1 - \cos {x}\), and the y-axis.
    1. Find the area of the region \( R \).
    2. Find the volume of the solid generated when the region \( R \) is revolved about the x-axis.
    3. The region \( R \) is the base of a solid. For this solid, each cross section perpendicular to the x-axis is a square. Find the volume of the solid.
  4. (2002 BC 1 Calc active) Let \(f\) and \(g\) be the functions given by \(f(x) = e^x\) and \(g(x) = \ln {x}\).
    1. Find the area of the region enclosed by the graphs of \(f\) and \(g\) between \(x = \frac{1}{2}\) and \(x = 1\).
    2. Find the volume of the solid generated when the region enclosed by the graphs of \(f\) and \(g\) between \(x = \frac{1}{2}\) and \(x = 1\) is revolved about the line \(y = 4\).
    3. Let \(h\) be the function given by \(h(x) = f(x) - g(x)\). Find the absolute minimum value of \(h(x)\) on the closed interval \(\frac{1}{2} \leq x \leq 1\), and find the absolute maximum value of \(h(x)\) on the closed interval \(\frac{1}{2} \leq x \leq 1\). Show the analysis that leads to your answers.