Grade 12 · Math · Calculus

12th grade Volumes by washers worksheet

6 printable versionsChallenge Me10 questions · answer keyFree · no account

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Volumes by washers

Grade 12 Math Worksheet - Challenge Me

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

The region bounded by y = √x, y = 0 and x = 1 spins about the line x = −1. Ari uses horizontal washers. Write the volume as V = kπ. What is k?
The region bounded by y = √x, y = 0 and x = 4 spins about the line x = −1. Bo uses horizontal washers. Write the volume as V = kπ. What is k?
The region bounded by y = √x, y = 0 and x = 9 spins about the line x = −1. Esme uses horizontal washers. Write the volume as V = kπ. What is k?
The region bounded by y = √x, y = 0 and x = 16 spins about the line x = −1. Jun uses horizontal washers. Write the volume as V = kπ. What is k?
Kira spins the region between y = 1 and y = sin x, 0 ≤ x ≤ π/2, about the x-axis. To the nearest hundredth, what is k in V = kπ?
Luz spins the region between y = 2 and y = cos x, 0 ≤ x ≤ π/2, about the x-axis. To the nearest hundredth, what is k in V = kπ?
Milo spins the region between y = sec x and y = 1, 0 ≤ x ≤ π/4, about the x-axis. To the nearest hundredth, what is k in V = kπ?
Noah spins the region between y = 2 and y = 1 + cos x, 0 ≤ x ≤ π, about the x-axis. To the nearest hundredth, what is k in V = kπ?
The region between y = R and y = 1, for 0 ≤ x ≤ 4, spins about the x-axis. Its volume is 32π. Pia solves for R > 1. What is R?
The region between y = R and y = 2, for 0 ≤ x ≤ 2, spins about the x-axis. Its volume is 24π. Ren solves for R > 2. What is R?

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

Standard: none (beyond CCSS)

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.

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.

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.

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