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IB Mathematics: Applications and Interpretation HL · AHL 3: Further Integration · AHL 3.9

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Volumes of revolution about the y-axis Problem set B

Sketch the region first. Write x² in terms of y, convert the limits to y-values, then use the calculator. Give answers to 3 significant figures unless told otherwise.

Each question matches the question with the same number in set A. Do the ones you got wrong there.

Difficulty: first skills applying them stretch

Skills

  1. The part of the curve \(y=3x^{2}+1\) for \(0\le x\le 2\) is rotated 360° about the y-axis. Find the volume of the solid enclosed by the curve, the y-axis and the horizontal lines through the two ends of the curve.

    0.511.522.52468101214xyy = 3x² + 1

  2. The part of the curve \(y=\ln x-1\) for \(2\le x\le 5\) is rotated 360° about the y-axis. Find the volume of the solid enclosed by the curve, the y-axis and the horizontal lines through the two ends of the curve.

    123456−1−0.50.511.5xy

  3. The part of the curve \(y=\cos x\) for \(0\le x\le 1.2\) is rotated 360° about the y-axis. Find the volume of the solid enclosed by the curve, the y-axis and the horizontal lines through the two ends of the curve. Use radians.

    0.20.40.60.811.21.40.20.40.60.811.2xyy = cos x

  4. The part of the curve \(y=\frac{3}{x-1}\) for \(2\le x\le 7\) is rotated 360° about the y-axis. Find the volume of the solid enclosed by the curve, the y-axis and the horizontal lines through the two ends of the curve.

    246812345xyy = 3/(x − 1)

  5. The diagram shows the region bounded by the curve \(y=\frac{3}{x}\), the y-axis and the lines \(y=2\) and \(y=5\).

    0.511.521234567xyy = 3/x
    1. (a) Find the area of the region.
    2. (b) The region is rotated 360° about the y-axis. Find the exact volume of the solid formed.

  6. The region between the curve \(y=\sqrt{x^{2}+9}\), the y-axis and the line \(y=5\) is shown in the diagram. It is rotated 360° about the y-axis. Find the volume of the solid, as a multiple of π.

    12345123456xyy = √(x² + 9)

In context

  1. A lamp shade is made by rotating a curve about the y-axis, with lengths in centimetres. The part of the curve \(y=\frac{x^{2}}{3}+1\) for \(0\le x\le 3\) is rotated 360° about the y-axis. Find the volume of the solid enclosed by the curve, the y-axis and the horizontal lines through the two ends of the curve.

    0.511.522.533.51234xyy = x²/3 + 1

  2. A horn is made by rotating a curve about the y-axis, with lengths in decimetres. The part of the curve \(y=1+\ln x\) for \(1\le x\le 6\) is rotated 360° about the y-axis. Find the volume of the solid enclosed by the curve, the y-axis and the horizontal lines through the two ends of the curve.

    12345670.511.522.533.5xy

  3. A plant pot is made by rotating the curve \(y=12\ln(x+1)\), for \(0\le y\le 24\), 360° about the y-axis. Lengths are in cm.

    1234567510152025x (cm)y (cm)
    1. (a) Find the capacity of the pot in litres.
    2. (b) The pot holds 400 cm³ of water. Find the depth of the water.

Stretch

  1. The diagram shows the region bounded by the curve \(x=6-(y-3)^{2}\) and the line \(x=2\). The region is rotated 360° about the y-axis. Find the volume of the solid, as a multiple of π.

    1234567123456xy

  2. A candle holder is made by rotating a curve about the y-axis. The part of the curve \(y=x^{2}+2\) for \(0\le x\le k\) is rotated 360° about the y-axis. The solid enclosed by the curve, the y-axis and the horizontal lines through the ends of the curve has volume 80 cm³. Find the value of k.

jmaths.xyz · AI HL · AHL 3.9 Volumes of revolution about the y-axis