PencilGo

Grade 12 · Math · Calculus

12th grade Volumes by washers worksheet

6 printable versionsOn Level10 questions · answer keyFree · no account

Level: Building Confidence · On Level · Challenge Me

When it’s done, scan the sheet with your phone and PencilGo marks it and picks the next one. Create a free account

Version
Print

Volumes by washers

Grade 12 Math Worksheet

NameDate

Circle the best answer, or write your answer on the line.

  1. The region between y = x² and y = 4 spins about the line y = −1. Ari uses washers. Write the volume as V = kπ. What is k?

  2. The region between y = x² and y = 4 spins about the line y = −2. Bo uses washers. Write the volume as V = kπ. What is k?

  3. The region between y = x² and y = 1 spins about the line y = −1. Esme uses washers. Write the volume as V = kπ. What is k?

  4. The region between y = x² and y = 9 spins about the line y = −1. Jun uses washers. Write the volume as V = kπ. What is k?

  5. The region between y = 5x and y = x² spins about the y-axis. Kira uses horizontal washers in y. Write the volume as V = kπ. What is k?

  6. The region between y = 6x and y = x² spins about the y-axis. Luz uses horizontal washers in y. Write the volume as V = kπ. What is k?

  7. The region between y = 8x and y = x² spins about the y-axis. Noah uses horizontal washers in y. Write the volume as V = kπ. What is k?

  8. The region between y = x and y = x³, x ≥ 0 spins about the x-axis. Pia sets up washers. Which integral gives the volume?

    • π ∫ from 0 to 1 of (x² − x⁶) dx
    • π ∫ from 0 to 1 of (x − x³)² dx
    • π ∫ from 0 to 1 of (x − x³) dx
    • π ∫ from 0 to 1 of (x⁶ − x²) dx

  9. The region between y = √x and y = x/3 spins about the x-axis. Tala sets up washers. Which integral gives the volume?

    • π ∫ from 0 to 9 of (x²/9 − x) dx
    • π ∫ from 0 to 9 of (√x − x/3)² dx
    • π ∫ from 0 to 9 of (√x − x/3) dx
    • π ∫ from 0 to 9 of (x − x²/9) dx

  10. The region between y = 9 and y = x² spins about the x-axis. Zee sets up washers. Which integral gives the volume?

    • π ∫ from −3 to 3 of (9 − x²)² dx
    • π ∫ from −3 to 3 of (x⁴ − 81) dx
    • π ∫ from −3 to 3 of (9 − x²) dx
    • π ∫ from −3 to 3 of (81 − x⁴) dx
PencilGopencilgo.com

12th grade Volumes by washers worksheet – printable
Print: 123456

About this worksheet

Volumes by washers. Every version has 10 questions, and the answer key prints on the second page.

What your child will learn

  • I can name the outer and inner radius of a washer.
  • I can find volumes by washers about the x-axis, the y-axis and other lines.
  • I can work back from a volume to a radius.

Standard: none (beyond CCSS)

Words to know

washer: A thin slice shaped like a coin with a hole.

outer radius R: Distance from the spin line to the farther curve.

inner radius r: Distance from the spin line to the nearer curve.

Common mistakes

Squared R − r. A washer's area is π(R² − r²): square each radius, then subtract.

Used R − r without squaring. A ring's area is π(R² − r²).

Used the nearer curve as the outer radius.

Questions parents ask

What is the most common mistake with volumes by washers?

Squared R − r. A washer's area is π(R² − r²): square each radius, then subtract.

Are these 12th grade math worksheets free to print?

Yes. All 6 printable versions at each level are free to print, with the answer key, and no account is needed.

More calculus

See all Calculus

Every sheet you scan goes into your child’s record: dated, marked and ready for the year-end portfolio. Homeschool records

Ready for easier practice?

Print a sheet, scan it when it’s done, and PencilGo picks the next one.

Create a free account