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Turning flat regions into volumes

Mathematics · Calculus & Analysis · ages 19-20

Name ______________________ Date ____________

Rotate the rectangle 0 <= x <= 2, 1 <= y <= 4 about the x-axis. Its volume is a multiple of π. Type that multiple.
Answer: ______________
You choose disks or washers for rotation about the x-axis. Which pairing of strip direction and integration variable is correct?
- Horizontal strips, integrating with dy
- Vertical strips, integrating with dx
- Vertical strips, integrating with dy
- Horizontal strips, integrating with dx
Rotate the region between y=√x and y=0, for 0 <= x <= 4, about the x-axis. Which integral gives its volume?
- 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 the rectangle 0 <= x <= 2, 0 <= y <= 3 about the x-axis gives a cylinder of radius 3.
Circle one: True False
Rotate the triangle bounded by y=2x, y=6-2x and y=0 about the x-axis. Which setup uses horizontal shells to avoid splitting at the peak?
- 2π times the integral from 0 to 3 of y(3-y) dy
- 2π times the integral from 0 to 3 of y(3+y) dy
- 2π times the integral from 0 to 3 of y(6-2y) dy
- 2π times the integral from 0 to 1.5 of y(3-y) dy
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=x and y=x², for 0 <= x <= 1, about the x-axis. Which radii describe her perpendicular slices?
- R=x² and r=x, using vertical strips
- R=x-x² and r=0, using vertical strips
- R=x and r=x², using vertical strips
Rotate the region 1 <= x <= 2, -3 <= y <= -x about the x-axis. Which washer radii are correct?
- Outer radius -3 and inner radius -x, using signed heights
- Outer radius x and inner radius 3, using endpoint distances
- Outer radius 3 and inner radius x, using endpoint distances
- Outer radius 3-x and inner radius 0, using the strip length
A student uses π(R-r)² for a washer's area, where R>r>0. What should change?
- Use the diameter difference, giving π[2(R-r)]² for the area.
- Square π too, giving π²(R-r)² as the washer's area.
- Subtract the squared radii: use π(R²-r²).
Rotate 0 <= y <= x, 0 <= x <= 2, about the y-axis. Shells give 2π∫_0^2 x² dx; washers give π∫_0^2 (4-y²) dy. What does evaluating both show?
- Shells give 8π/3, while washers give 16π/3
- Shells give 16π/3, while washers give 8π/3
- Both give 8π/3 for the volume of this solid
- Both give 16π/3 for the volume of this solid

Answer key

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

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30 · Each washer has area π(4²-1²)=15π. Multiplying by the interval length 2 gives volume 30π.
Vertical strips, integrating with dx · Disks and washers use strips perpendicular to the axis. The x-axis is horizontal, so those strips are vertical and their thickness is measured in x.
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 perpendicular cross-section is a disk of radius 3. The solid is a cylinder of radius 3 and length 2.
2π times the integral from 0 to 3 of y(3-y) dy · At height y, left x=y/2 and right x=3-y/2. Their difference is 3-y. The shell radius is y and the peak height is 3.
8 · The integral of x from 0 to 4 is 8, so the volume is 8 pi.
R=x and r=x², using vertical strips · On [0,1], x>=x²>=0. The farther boundary gives outer radius x, and the nearer boundary gives inner radius x².
Outer radius 3 and inner radius x, using endpoint distances · Distances from the x-axis are |-3|=3 and |-x|=x. Since 1<=x<=2<3, the outer radius is 3 and the inner radius is x.
Subtract the squared radii: use π(R²-r²). · The washer is an outer circle with an inner circle removed. Its area is πR²-πr², not the area of a circle of radius R-r.
Both give 16π/3 for the volume of this solid · Shells give 2π(8/3)=16π/3. Washers give π(8-8/3)=16π/3. The two valid setups agree.