Cylindrical coordinates

When to choose this method

• The horizontal projection is circular and height is described by z and x²+y².
• Cylindrical coordinates retain z and transform only the planar coordinates.

Before calculating

1. Determine the projection and the lower and upper surfaces.
2. Choose an order compatible with dependencies. The calculator integrates dr, dθ, dz in cylindrical coordinates.

A complete example, step by step

1. The inequality requires z≥0 and r²≤4. The projection is a disk of radius 2.

   $$0\le r\le2,\quad0\le z\le4-r^2$$

2. We can also slice at height z. Solve for the maximum radius; z is now the outer variable.

   $$0\le z\le4,\quad0\le r\le\sqrt{4-z},\quad0\le\theta\le2\pi$$

3. The volume integrand is 1; the Jacobian supplies r. Use the editor-supported order.

   $$V=\int_0^4\int_0^{2\pi}\int_0^{\sqrt{4-z}}r\,dr\,d\theta\,dz$$

4. Integrate radially while treating z as constant.

   $$\int_0^{\sqrt{4-z}}r\,dr=\frac{4-z}{2}$$

5. The angular integral gives 2π.

   $$V=\pi\int_0^4(4-z)\,dz$$

6. Evaluate height from 0 to 4 to obtain the volume.

   $$V=\pi\left[4z-\frac{z^2}{2}\right]_0^4=8\pi$$

Now try it yourself

Solve on paper before opening the hint or answer. These are self-assessment activities; to check a typed answer, open Learn.

In these exercises θ runs from 0 to 2π: a full revolution is intended.

Connect with other methods