Volumes by Disk and Washer Methods Practice

Sample questions

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1. Sample 1difficulty 2/5

   For $y = c$ (constant) revolved on $[a, b]$ about $x$-axis: Volume is

   • A

     $\pi c (b - a)$

   • B

     $c (b - a)$

   • C

     $\pi c^2 (b - a)$

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   • D

     $2\pi c (b - a)$

   Why

   Cylinder volume: $\pi r^2 h = \pi c^2 (b - a)$.

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2. Sample 2difficulty 2/5

   For revolution about the $x$-axis, the radius of the disk at location $x$ is

   • A

     $f(x)$

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   • B

     $x^2$

   • C

     $x$

   • D

     $f'(x)$

   Why

   Distance from axis to curve at $x$ is $f(x)$.

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3. Sample 3difficulty 3/5

   Cone of radius $R$, height $H$ via $y = (R/H) x$, $0 \le x \le H$, revolved:

   • A

     $\tfrac{2}{3}\pi R^2 H$

   • B

     $\tfrac{1}{3}\pi R^2 H$

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   • C

     $\pi R^2 H$

   • D

     $\tfrac{1}{2}\pi R^2 H$

   Why

   Standard cone volume formula confirmed by integration.

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4. Sample 4difficulty 3/5

   The disk method computes a volume of revolution around the $x$-axis as

   • A

     $\int_a^b 2\pi f(x)\,dx$

   • B

     $\int_a^b f(x)\,dx$

   • C

     $\int_a^b \pi [f(x)]^2\,dx$

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   • D

     $\int_a^b 2\pi x\,f(x)\,dx$

   Why

   Each disk has radius $f(x)$ and thickness $dx$ → volume $\pi[f(x)]^2\,dx$.

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5. Sample 5difficulty 3/5

   Rotate $x = y^2$, $0 \le y \le 2$, about the $y$-axis. Volume by disks (in $y$):

   • A

     $\pi\int_0^2 y^2\,dy = \dfrac{8\pi}{3}$

   • B

     $\int_0^2 y^4\,dy$

   • C

     $2\pi\int_0^2 y^2\,dy$

   • D

     $\pi\int_0^2 y^4\,dy = \dfrac{32\pi}{5}$

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   Why

   Disks of radius $x = y^2$, thickness $dy$: $\pi y^4\,dy$.

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