Problem 10 - Calculus Early Transcendentals

Question

What coordinate system is suggested if the integrand of a triple integral involves \(x^{2}+y^{2}+z^{2} ?\)

Step-by-Step Solution

Answer

Answer: Spherical Coordinates (\(\rho, \theta, \phi\)).

Step 1: Evaluate the integrand

The integrand involves the sum of squares of the variables: \(x^2+y^2+z^2\). It's worth noting that this expression embodies the square of the euclidean distance from the origin to the point (x, y, z) in a Cartesian coordinate system.

Step 2: Compare with coordinate systems

Now, let's recall the main features and formulas for each coordinate system: 1. Cartesian Coordinates: \((x, y, z)\) 2. Cylindrical Coordinates: \((r, \theta, z)\) where \(r = \sqrt{x^2 + y^2}\), \(x = r\cos\theta\), \(y = r\sin\theta\), and \(z = z\). 3. Spherical Coordinates: \((\rho, \theta, \phi)\) where \(\rho = \sqrt{x^2 + y^2 + z^2}\), \(x = \rho\sin\phi\cos\theta\), \(y = \rho\sin\phi\sin\theta\), and \(z = \rho\cos\phi\).

Step 3: Identify the suitable coordinate system

Notice that the integrand \(x^2 + y^2 + z^2\) is equivalent to \(\rho^2\) in spherical coordinates. This means that spherical coordinates are likely a more natural choice, as the integrand simplifies to a single term. In cylindrical coordinates, the integrand becomes \(r^2 + z^2\), which is still a sum of squares. This is less suitable than spherical coordinates, but potentially more suitable than Cartesian coordinates.

Step 4: Conclusion

Based on the integrand \(x^2 + y^2 + z^2\), the suggested coordinate system would be Spherical Coordinates, represented by \((\rho, \theta, \phi)\).