
# 9.5: Triple Integrals in Cylindrical and Spherical Coordinates


Earlier in this chapter we showed how to convert a double integral in rectangular coordinates into a double integral in polar coordinates in order to deal more conveniently with problems involving circular symmetry. A similar situation occurs with triple integrals, but here we need to distinguish between cylindrical symmetry and spherical symmetry. In this section we convert triple integrals in rectangular coordinates into a triple integral in either cylindrical or spherical coordinates.

Also recall the chapter prelude, which showed the opera house l’Hemisphèric in Valencia, Spain. It has four sections with one of the sections being a theater in a five-story-high sphere (ball) under an oval roof as long as a football field. Inside is an IMAX screen that changes the sphere into a planetarium with a sky full of $$9000$$ twinkling stars. Using triple integrals in spherical coordinates, we can find the volumes of different geometric shapes like these.

## Review of Cylindrical Coordinates

As we have seen earlier, in two-dimensional space $$\mathbb{R}^2$$ a point with rectangular coordinates $$(x,y)$$ can be identified with $$(r,\theta)$$ in polar coordinates and vice versa, where $$x = r \, \cos \theta$$, $$y = r \, \sin \, \theta, \, r^2 = x^2 + y^2$$ and $$\tan \, \theta = \left(\frac{y}{x}\right)$$ are the relationships between the variables.

In three-dimensional space $$\mathbb{R}^3$$ a point with rectangular coordinates $$(x,y,z)$$ can be identified with cylindrical coordinates $$(r, \theta, z)$$ and vice versa. We can use these same conversion relationships, adding $$z$$ as the vertical distance to the point from the $$(xy$$-plane as shown in $$\PageIndex{1}$$.

To convert from rectangular to cylindrical coordinates, we use the conversion

• $$x = r \, \cos \theta$$
• $$y = r \, \sin \, \theta$$
• $$z=z$$

To convert from cylindrical to rectangular coordinates, we use

• $$r^2 = x^2 + y^2$$ and
• $$\theta = \tan^{-1} \left(\frac{y}{x}\right)$$
• $$z=z$$

Note that that $$z$$-coordinate remains the same in both cases.

In the two-dimensional plane with a rectangular coordinate system, when we say $$x = k$$ (constant) we mean an unbounded vertical line parallel to the $$y$$-axis and when $$y = l$$ (constant) we mean an unbounded horizontal line parallel to the $$x$$-axis. With the polar coordinate system, when we say $$r = c$$ (constant), we mean a circle of radius $$c$$ units and when $$\theta = \alpha$$ (constant) we mean an infinite ray making an angle $$\alpha$$ with the positive $$x$$-axis.

Similarly, in three-dimensional space with rectangular coordinates $$(x,y,z)$$ the equations $$x = k, \, y = l$$ and $$z = m$$ where $$k, \, l$$ and $$m$$ are constants, represent unbounded planes parallel to the $$yz$$-plane, $$xz$$-plane and $$xy$$-plane, respectively. With cylindrical coordinates $$(r, \theta, z)$$, by $$r = c, \, \theta = \alpha$$, and $$z = m$$, where $$c, \alpha$$, and $$m$$ are constants, we mean an unbounded vertical cylinder with the

## Integration in Spherical Coordinates

feet. The bottom of the balloon is modeled by a frustum of a cone (think of an ice cream cone with the pointy end cut off).

The radius of the large end of the frustum is $$28$$ feet and the radius of the small end of the frustum is $$28$$ feet. A graph of our balloon model and a cross-sectional diagram showing the dimensions are shown in the following figure.

## Key Concepts

#### Glossary

9.5: Triple Integrals in Cylindrical and Spherical Coordinates is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.