Write a triple integral including limits of integration that gives the volume of the cap of the solid sphere x2+y2+z2≤5 cut off by the plane z=2 and restricted to the first octant. (in your integral, use theta, rho, and phi for θ, ρ and ϕ, as needed.)

Since it is in the first octant, if we let be the angle formed in the xy plane, then its limits are simply .

The radius is a constant of so we integrate from [tex0 \leq \rho \leq \sqrt 5[/tex]

The yz/xz plane doesn't go the full 90 degrees. Instead, it goes to the plane which means that it forms a triangle of hypotenuse and an opposite leg of . This produces an angle of so our limits for

We are just integrating a constant of 1, then we get:

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