13. Volume

Homework

  1. The base of a solid is the area between the semicircle \(y=\sqrt{9-x^2}\) and the \(x\)-axis. Find its volume if the cross-sections perpendicular to the \(x\)-axis are squares.

    This animation shows a solid whose base is a semicircle with diameter along the x axis with square cross sections perpendicular to the x axis. The cross sections are accumulating to fill in the solid.

  2. The base of a solid is the region between \(x=2y^2\) and \(x=8y\). The cross-sections perpendicular to the \(y\)-axis are semicircles. Find the volume.

    This figure shows the region to the right of the parabola y = 2 x squared and to the left of the line x = 8y. There is a horizontal line between the parabola and the line.

  3. In each problem, the planar region bounded by the given curves is rotated about the indicated line to sweep out a volume. Will you do an \(x\)-integral or a \(y\)-integral? Are the Riemann sum rectangles horizontal or vertical? When the rectangle is rotated, does it sweep out a thin disk, thin washer or thin cylinder?

    1. Between \(y=x^2\) and \(y=8-x^2\) about the \(x\)-axis.

    2. Between \(x=y^3\) and the \(y\)-axis, between \(y=1\) and \(y=3\) about the \(y\)-axis.

    3. Between \(x=y^2+2\) and \(x=6\) for \(y \ge 0\) about the \(x\)-axis.

    4. Between \(x=y^2+2\) and \(x=6\) for \(y \ge 0\) about the line \(y=-2\).

    5. Between \(y=x^2\) and \(y=4\) about the line \(x=2\).

    6. Between \(y=x^2\) and \(y=4\) about the line \(y=8\).

    7. Between \(y=\sqrt{x}\) and the \(x\)-axis, between \(x=0\) and \(x=4\) about the \(y\)-axis.

    8. Between \(y=\sqrt{x}\) and the \(x\)-axis, between \(x=0\) and \(x=4\) about the \(x\)-axis.

    9. Between \(x=y^2+2\) and \(x=6\) for \(y \ge 0\) about the \(y\)-axis.

    10. Between \(x=y^2+2\) and \(x=6\) for \(y \ge 0\) about the line \(x=8\).

    11. Between \(x=5-y^2\) and the line \(x=1\) about the line \(x=1\).

    12. Between \(y=x^2\) and \(y=4\) about the line \(y=4\).

  4. The area between \(y=\sqrt{x}\) and the \(x\)-axis for \(0 \le x \le 4\) is rotated about the \(x\)-axis. Find the volume swept out.

  5. The area between \(y=\sqrt{x}\) and the \(x\)-axis for \(0 \le x \le 4\) is rotated about the \(y\)-axis. Find the volume swept out.

  6. The area between \(y=\sqrt{x}\) and \(y=x\) is rotated about the \(x\)-axis. Find the volume swept out.

  7. The area between \(y=\sqrt{x}\) and \(y=x\) is rotated about the \(y\)-axis. Find the volume swept out.

  8. The area between \(x=y^2+2\) and \(x=6\) for \(y \ge 0\) is rotated about the \(x\)-axis. Find the volume swept out.

  9. The area between \(x=y^2+2\) and \(x=6\) for \(y \ge 0\) is rotated about the line \(x=8\). Find the volume swept out.

  10. The area between \(y=e^x\) and the \(x\)-axis for \(0 \le x \le 1\) is rotated about the line \(y=-1\). Find the volume swept out.

  11. The area between \(y=e^x\) and the \(x\)-axis for \(0 \le x \le 1\) is rotated about the \(y\)-axis. Find the volume swept out.

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Supported in part by NSF Grant #1123255

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