Volume

Learn disk volumes, washer volumes.

Lesson 2 of 32 parts Practice Flashcards

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Disk Volumes

Objective: Explain disk volumes, apply the stated method, and check its conditions.

Rotating a region touching its axis of revolution gives disk cross-sections with area πR². Integrate these cross-sectional areas along the axis. The radius is measured from the rotation axis to the curve and volume is expressed in cubic units. Use radius squared, not diameter squared.

Formulas

$V=π∫R(x)²dx$

Worked examples

Try itRotate y=6 on 0≤x≤7 around the x-axis. Give the solid volume as a coefficient of π.Show answer

Step 1: The disk radius is 6. Step 2: V=π∫₀^7 36 dx. Step 3: V=252π cubic units.

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Disk Volumes

Objective: Explain disk volumes, apply the stated method, and check its conditions.

Formulas

$V=π∫R(x)²dx$

Worked examples

Try itRotate y=6 on 0≤x≤7 around the x-axis. Give the solid volume as a coefficient of π.Show answer

Step 1: The disk radius is 6. Step 2: V=π∫₀^7 36 dx. Step 3: V=252π cubic units.

Washer Volumes

Objective: Explain washer volumes, apply the stated method, and check its conditions.

A rotated region separated from the axis gives washer cross-sections of area π(R²−r²), where R≥r≥0. Subtract inner disk area from outer disk area before integrating. Squaring the radius difference is not equivalent to subtracting squared radii. R²−r² is not generally (R−r)².

Formulas

$V=π∫(R²−r²)dx$

Worked examples

Try itRotate the region between y=13 and y=6 on [0,7] about the x-axis. Give volume as a coefficient of π.Show answer

Step 1: Outer radius R=13; inner radius r=6. Step 2: V=π∫₀^7(R²−r²) dx. Step 3: The coefficient is 931.

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