riaology
Unit 5 · Lesson 3 of 4

A strip parallel to the axis becomes a shell.

A thin vertical strip can rotate into a hollow cylindrical layer. Its volume comes from circumference, height, and radial thickness.

Change the input. Follow the mathematics.

What you'll be able to do
  • Set up the model and calculate its solid volume.
  • Connect the equation to the changing visual.
  • Explain the assumptions and identify a misleading interpretation.
Predict before exploring

What will change if you increase upper radius b? Predict the direction, then explain the response.

A model you can changeMove one control at a time.
Solid of revolution illustrated with a cross-section and a thin ringCircumference × height × thickness
Radius x (m) versus Shell volume density (m²). Shell contribution per dx. Selected solid volume: 2.0944 m³. Shell radius, height, and thickness are separate factors: circumference × height × dx. Strips are parallel to the rotation axis.003.1420.56.2831Shell volume density (m²)Radius x (m)
Shell contribution per dx

solid volumeV=2π(1)33=2.0944 m3\mathcal V=\frac{2\pi(1)^3}{3}=2.0944\,\mathrm{m^3}

Shell radius, height, and thickness are separate factors: circumference×height×dxcircumference \times height \times dx. Strips are parallel to the rotation axis.

Assumptions, units, and limits

Rotate the region under y=xy=x on 0≤x≤b about the y axis. Each vertical strip becomes a cylindrical shell.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

01

From the situation to an equation

Choose strips parallel to the rotation axis. Their volume is approximately circumference times strip height times thickness. Integrate over radius; a different axis changes the radius expression.

V=∫0b2πx x dx=2πb33\mathcal V=\int_0^b2\pi x\,x\,dx=\frac{2\pi b^3}3
02

Keep the reasoning visible

dV=2π(radius)(height) dxd\mathcal V=2\pi(\text{radius})(\text{height})\,dx

Shell radius, height, and thickness are separate factors: circumference×height×dxcircumference \times height \times dx. Strips are parallel to the rotation axis.

03

What makes this model useful?

Rotate the region under y=xy=x on 0≤x≤b about the y axis. Each vertical strip becomes a cylindrical shell.

Shell height is the difference of bounding curves, not automatically the distance to the axis. Avoid counting a region twice.

A worked example

Calculate the starting scenario

  1. Use upper radius b = 1 m.
  2. V=2π(1)33=2.0944 m3\mathcal V=\frac{2\pi(1)^3}{3}=2.0944\,\mathrm{m^3}
  3. Shell radius, height, and thickness are separate factors: circumference×height×dxcircumference \times height \times dx. Strips are parallel to the rotation axis. The checked quantity is solid volume.
Make the idea your own

Calculate. Compare. Explain.

Numeric checks allow 1% rounding error. Values near zero use a small absolute tolerance. Work through the setup before opening a solution.

Calculate and explain

Rebuild the starting calculation

Calculate the solid volume using upper radius b 1 m. State the meaning of its sign and units.

Checked quantity: solid volume · m³. Follow the definition for finite-cutoff and numerical-model questions.

Hint

Choose strips parallel to the rotation axis. Their volume is approximately circumference times strip height times thickness. Integrate over radius; a different axis changes the radius expression.

Explore the visual solution
Solid of revolution illustrated with a cross-section and a thin ringCircumference × height × thickness
Radius x (m) versus Shell volume density (m²). Shell contribution per dx. Selected solid volume: 2.0944 m³. Shell radius, height, and thickness are separate factors: circumference × height × dx. Strips are parallel to the rotation axis.003.1420.56.2831Shell volume density (m²)Radius x (m)
Shell contribution per dx

solid volumeV=2π(1)33=2.0944 m3\mathcal V=\frac{2\pi(1)^3}{3}=2.0944\,\mathrm{m^3}

Shell radius, height, and thickness are separate factors: circumference×height×dxcircumference \times height \times dx. Strips are parallel to the rotation axis.

Assumptions, units, and limits

Rotate the region under y=xy=x on 0≤x≤b about the y axis. Each vertical strip becomes a cylindrical shell.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

The controls explore variations. These written steps retain the original question's inputs.

Choose strips parallel to the rotation axis. Their volume is approximately circumference times strip height times thickness. Integrate over radius; a different axis changes the radius expression.

V=∫0b2πx x dx=2πb33\mathcal V=\int_0^b2\pi x\,x\,dx=\frac{2\pi b^3}3
Calculate and explain

Apply the idea to a new scenario

For region under y=xy=x on 0≤x≤1.5, find the volume about the y axis using shells.

Checked quantity: solid volume · m³. Follow the definition for finite-cutoff and numerical-model questions.

Hint

Choose strips parallel to the rotation axis. Their volume is approximately circumference times strip height times thickness. Integrate over radius; a different axis changes the radius expression.

Explore the visual solution
Solid of revolution illustrated with a cross-section and a thin ringCircumference × height × thickness
Radius x (m) versus Shell volume density (m²). Shell contribution per dx. Selected solid volume: 7.06858 m³. Shell radius, height, and thickness are separate factors: circumference × height × dx. Strips are parallel to the rotation axis.007.0690.7514.1371.5Shell volume density (m²)Radius x (m)
Shell contribution per dx

solid volumeV=2π(1.5)33=7.06858 m3\mathcal V=\frac{2\pi(1.5)^3}{3}=7.06858\,\mathrm{m^3}

Shell radius, height, and thickness are separate factors: circumference×height×dxcircumference \times height \times dx. Strips are parallel to the rotation axis.

Assumptions, units, and limits

Rotate the region under y=xy=x on 0≤x≤b about the y axis. Each vertical strip becomes a cylindrical shell.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

The controls explore variations. These written steps retain the original question's inputs.

Choose strips parallel to the rotation axis. Their volume is approximately circumference times strip height times thickness. Integrate over radius; a different axis changes the radius expression.

V=∫0b2πx x dx=2πb33\mathcal V=\int_0^b2\pi x\,x\,dx=\frac{2\pi b^3}3
Challenge the interpretation

Which explanation survives the model?

Choose the statement you can justify, then use the visual to explain why the others fail.

Test it with the visual
Solid of revolution illustrated with a cross-section and a thin ringCircumference × height × thickness
Radius x (m) versus Shell volume density (m²). Shell contribution per dx. Selected solid volume: 2.0944 m³. Shell radius, height, and thickness are separate factors: circumference × height × dx. Strips are parallel to the rotation axis.003.1420.56.2831Shell volume density (m²)Radius x (m)
Shell contribution per dx

solid volumeV=2π(1)33=2.0944 m3\mathcal V=\frac{2\pi(1)^3}{3}=2.0944\,\mathrm{m^3}

Shell radius, height, and thickness are separate factors: circumference×height×dxcircumference \times height \times dx. Strips are parallel to the rotation axis.

Assumptions, units, and limits

Rotate the region under y=xy=x on 0≤x≤b about the y axis. Each vertical strip becomes a cylindrical shell.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

The reasoning. Choose strips parallel to the rotation axis. Their volume is approximately circumference times strip height times thickness. Integrate over radius; a different axis changes the radius expression.

Your learning path

Can you explain the connection?

Mark complete when you can set up the calculation and explain what the visual is showing. Opening a page never completes it automatically.

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