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Cogito

Multivariable Calculus · Chapter 7 · Lesson 3

Practice Worksheet

Triple Integrals

Three integrals, and the answer is mass or volume.

12 problems · about 24 minutes · F-IF.B.6, G-GMD.A.3

NameDate

Figure — use these to answer the problems

A cone is the natural solid for cylindrical coordinates. Its height depends only on the distance from the axis.
  • The surfacez = a·√(x² + y²)
  • Height at the origin0
a, the steepness1
b1
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. A solid of volume 9 with constant density 6. What is its mass?

    Answer

  2. What does integrating the constant 1 over a solid give?

    • The volume of the solid.
    • The surface area of the solid.

  3. A box 2 by 2 by 2. What is its volume?

    Answer

  4. A solid of volume 12 with constant density 7. What is its mass?

    Answer

  5. How many nested integrals does a triple integral have?

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. Must the outermost limits be plain numbers? 1 yes, 0 no.

    Answer

  2. The cylindrical volume element carries r to what power?

    Answer

  3. A cylinder of radius 2 and height 5. What is its volume divided by π?

    Answer

Stretch Yourself

Mixed problems. Work out what kind of question it is before you start.

4 problems

  1. Match each integral to what it computes.

    Draw a line from each item on the left to its match on the right.

    • Triple integral of 1
    • Triple integral of a density
    • Double integral of 1
    • Volume of a solid
    • Mass of a solid
    • Area of a region

  2. A cube of side 4. What is its volume?

    Answer

  3. The Block

    A block 3 by 4 by 5 with constant density 2. What is its mass?

    Answer

  4. The Volume

    A triple integral of the constant 1 over a solid gives 45. What is the volume?

    Answer