To set up the integral for the volume of the cap of the sphere in spherical coordinates, let's go through each part:

Step 1: Recognize Given Information

The equation of the sphere is $x^2 + y^2 + z^2 = 34$, which gives the radius $\rho = \sqrt{34}$ in spherical coordinates. The sphere is cut by the plane $z = 5$, and we are only interested in the portion within the first octant.

Step 2: Choose Coordinate System

Since we're dealing with a sphere and have a cap defined by a plane, spherical coordinates are appropriate.

Step 3: Determine Limits for Spherical Coordinates

1. For $\rho$:

   • $\rho$ ranges from 0 to the radius of the sphere but is limited by the plane $z = 5$.
   • In spherical coordinates, $z = \rho \cos \phi$.
   • Set $\rho \cos \phi = 5$ to find the limit for $\rho$, which gives $\rho = \frac{5}{\cos \phi}$.
   • Therefore, $\rho$ ranges from $0$ to $\min(\sqrt{34}, \frac{5}{\cos \phi})$.

2. For $\theta$:

   • Since we are in the first octant, $\theta$ (azimuthal angle) ranges from $0$ to $\frac{\pi}{2}$.

3. For $\phi$:

   • $\phi$ (polar angle) is restricted to the portion above the plane $z = 5$.
   • Given $\cos \phi = \frac{5}{\sqrt{34}}$, $\phi$ ranges from $0$ to $\cos^{-1} \left(\frac{5}{\sqrt{34}}\right)$.

Step 4: Integrate in Spherical Coordinates

The volume element in spherical coordinates is $dV = \rho^2 \sin \phi \, d\rho \, d\phi \, d\theta$.

Filling in the Blanks

1. Coordinates used: spherical
2. Limits of Integration:
   • $a = 0$, $b = \frac{\pi}{2}$ (for $\theta$)
   • $c = 0$, $d = \cos^{-1} \left(\frac{5}{\sqrt{34}}\right)$ (for $\phi$)
   • $e = 0$, $f = \frac{5}{\cos \phi}$ (for $\rho$)
3. Integral Expression: $\text{Volume} = \int_0^{\frac{\pi}{2}} \int_0^{\cos^{-1} \left(\frac{5}{\sqrt{34}}\right)} \int_0^{\frac{5}{\cos \phi}} \rho^2 \sin \phi \, d\rho \, d\phi \, d\theta$

Would you like further details on the integration process, or do you have questions about specific steps?

1. How do spherical coordinates simplify triple integrals involving spheres?
2. Why does the plane $z = 5$ change the upper limit for $\rho$?
3. What role does the first octant restriction play in setting the limits?
4. How would the limits change if it weren’t restricted to the first octant?
5. What other coordinate systems could be used, and how would that affect the setup?

Tip: When setting up triple integrals in spherical coordinates, always focus on finding the radial, polar, and azimuthal limits separately based on geometric constraints.