Lesson Worksheet: Volumes of Solids of Revolution Using Disk and Washer Method Mathematics

In this worksheet, we will practice finding the volume of a solid generated by revolving a region around either a horizontal or a vertical line using integration.

Q1:

Find the volume of the solid obtained by rotating the region bounded by the curves 𝑦=4+𝑥sec and 𝑦=6 about 𝑦=4 where 𝑥𝜋2,𝜋2. Give your answer to two decimal places.

Q2:

Consider the region bounded by the curve 𝑦=34𝑥cos and the lines 𝑦=0, 𝑥=𝜋8, and 𝑥=𝜋8. Set up an integral for the volume of the solid obtained by rotating that region about 𝑦=4.

  • A𝜋484𝑥184𝑥𝑥coscosd
  • B𝜋64𝑥𝑥cosd
  • C𝜋244𝑥94𝑥𝑥coscosd
  • D𝜋34𝑥𝑥cosd
  • E𝜋94𝑥𝑥cosd

Q3:

Set up an integral for the volume of the solid obtained by rotating the region bounded by the curve 𝑦=𝑒 and the lines 𝑦=0, 𝑥=5, and 𝑥=5 about 𝑦=5.

  • A𝜋𝑒25𝑥d
  • B2𝜋𝑒+10𝑒𝑥d
  • C𝜋𝑒+25𝑥d
  • D𝜋𝑒+25𝑥d
  • E𝜋𝑒+10𝑒𝑥d

Q4:

Find the volume of the solid obtained by rotating the region bounded by the curves 𝑦=𝑥sin, 𝑦=𝑥cos, 𝑥=𝜋6, and 𝑥=𝜋4 about 𝑦=1. Give your answer to two decimal places.

Q5:

Find the volume of the solid obtained by rotating the region bounded by the curves 𝑦=8𝑥 and 𝑥=𝑦 about 𝑦=5.

  • A191𝜋480
  • B3𝜋80
  • C191𝜋240
  • D3𝜋160
  • E191𝜋960

Q6:

Set up an integral for the volume of the solid obtained by rotating the region bounded by the curve 9𝑥+𝑦=9 about 𝑦=5.

  • A301𝑥𝑥d
  • B30𝜋1𝑥𝑥d
  • C15𝜋1𝑥𝑥d
  • D60𝜋1𝑥𝑥d
  • E601𝑥𝑥d

Q7:

Consider the region bounded by the curves 𝑦=𝑥, 𝑦=0, and 𝑥=2. Find the volume of the solid obtained by rotating this region about 𝑥=3.

  • A96𝜋5
  • B64𝜋5
  • C112𝜋5
  • D128𝜋5
  • E56𝜋5

Q8:

Determine, to two decimal places, the volume of the solid obtained by rotating the region bounded by the curves 𝑥=5𝑦 and 𝑥=2𝑦 about 𝑥=3.

Q9:

Set up an integral for the volume of the solid obtained by rotating the region bounded by the curve 4𝑥+𝑦=4 about 𝑥=2.

  • A8𝜋1𝑦4𝑦d
  • B161𝑦4𝑦d
  • C16𝜋1𝑦4𝑦d
  • D81𝑦4𝑦d
  • E4𝜋1𝑦4𝑦d

This lesson includes 81 additional question variations for subscribers.

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