Is this the correct parametrization for this surface?

[ember fjord]

This question send to be missing so much information. What is the height of the prism? Where does the prism end and the tube begin?

Oh, is this translated from another language?

Okay, this looks mostly good. It's been a while for me, but why sin phi in the first integral? Actually, you can tell it's not quite right by checking it against the formula for the surface area of a sphere.

You could parameterize the half sphere the same way you do the tube, but polar is definitely more elegant. You just need the right equation.

And I think you're missing the underside of the half sphere where it's wider than the tube.

And part of the top face of the cube is covered by the tube.

[brazen charm]

I do have a cross section of R here:

[brazen charm]

So I could take the surface area of the cube and subtract by a circle with radius e^0 = 1, and take the surface area of the half-sphere, add 16π (circle radius 4, flat half) to it and subtract by a circle of radius e^1?

[brazen charm]

sin(phi) is the "area scaling element" like the one in the second integral (determinant of the cross product of ∂r/∂(theta) and ∂r/∂(phi) where r is the spherical parameterization of the sphere). I did check it against the formula for the surface area of a sphere by changing the bound of phi to π instead of just π/2 and it gave me the same result 😅

[cunning pilot]

Hey man can I ask a question to you?

[brazen charm]

.solved