Turning flat regions into volumes

Mathematics · Calculus & Analysis · ages 19-20

Name ______________________ Date ____________

A washer has outer radius 5 cm and inner radius 3 cm. What is outer radius squared minus inner radius squared?
Answer: ______________
The region under y = sqrt x from 0 to 4 is rotated about the x-axis. Which integral gives the volume?
- pi times the integral from 0 to 4 of x dx
- pi times the integral from 0 to 4 of sqrt x dx
- 2 pi times the integral from 0 to 4 of x dx
- pi times the integral from 0 to 4 of x squared dx
Which integral gives the volume from rotating y equals square root of x, 0 to 4, about the x-axis?
- pi times the integral from 0 to 4 of x dx
- 2 pi times the integral from 0 to 4 of x dx
- pi times the integral from 0 to 4 of square root of x dx
Rotating y equals 3 from 0 to 2 about the x-axis gives a cylinder of radius 3.
Circle one: True False
The region under y = x squared from 0 to 1 is rotated about the y-axis. Why do shells give the simpler integral?
- Vertical strips need no rewriting: radius x and height x squared with dx
- Horizontal strips avoid all square roots
- Shells remove pi from the formula
- Disks cannot be used for regions touching an axis
Pi times the integral from 0 to 4 of x dx equals some multiple of pi. Type that multiple.
Answer: ______________
Maya rotates the region between y equals x and y equals x squared, 0 to 1, about the x-axis. Which describes her washers?
- Outer radius x squared, inner radius x
- A single full disk of radius x minus x squared
- Outer radius x, inner radius x squared
The region between y = x on top and y = x squared below, from 0 to 1, is rotated about the x-axis. Which describes the washers?
- Outer radius x, inner radius x squared
- Outer radius x squared, inner radius x
- Outer radius 1, inner radius x minus x squared
- A single disk of radius x minus x squared
A student writes a washer volume as pi times the integral of outer minus inner, squared. What is the mistake?
- Bounds must be squared as well
- Pi must be squared too
- Each radius must be squared first, then subtracted
With shells, which integral gives the volume for the region under y = x squared from 0 to 1 rotated about the y-axis?
- 2 pi times the integral from 0 to 1 of x cubed dx
- pi times the integral from 0 to 1 of x to the fourth dx
- 2 pi times the integral from 0 to 1 of x squared dx
- pi times the integral from 0 to 1 of x dx

Answer key

For grown-ups. Fold this page away before handing over the rest.

Turning flat regions into volumes W1-mt_mzZ22MkWD4-s1

16 · 25 minus 9 is 16, and the washer area is 16 pi square cm.
pi times the integral from 0 to 4 of x dx · Disks have radius sqrt x, and radius squared is x, so the volume is pi times the integral of x from 0 to 4.
pi times the integral from 0 to 4 of x dx · Disks have radius square root of x, and squaring the radius gives x.
True · Every strip has length 3, so every disk has radius 3.
Vertical strips need no rewriting: radius x and height x squared with dx · Shells keep x as the variable with height x squared, while disks would force rewriting the curve as x equals sqrt y.
8 · The integral of x from 0 to 4 is 8, so the volume is 8 pi.
Outer radius x, inner radius x squared · The top curve gives the outer edge and the bottom curve gives the inner edge.
Outer radius x, inner radius x squared · The hole comes from the lower curve, so the outer edge follows y equals x and the inner edge follows y equals x squared.
Each radius must be squared first, then subtracted · Washer area is pi times outer squared minus inner squared.
2 pi times the integral from 0 to 1 of x cubed dx · Shell radius is x and height is x squared, so 2 pi r h is 2 pi x cubed.