Height is the density, the floor is the region, the volume is the probability.
Setup: one slice of the solid at a fixed y. Pause: let the viewer predict first.
A(y) = 1/2 + y: the integrand's y is a constant while x moves.
Pair the integral signs with the differentials from the inside out, like brackets.
Pause: let the viewer predict first.
A triangle with corners (0,0), (0,2), (2,2): above the diagonal.
The inner integral is the length of one strip; the outer integral stacks them.
Pause: let the viewer predict first.
It enters on the diagonal y = x and leaves at the top edge y = 2.
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No: below y = 1 the strip ends on the diagonal, above it ends on the wall.
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The triangle with corners (0,0), (1,0), (0,1).
Inner integral with x frozen, factor, then the outer integral.
Pause: let the viewer predict first.
Because the density is small near (0,0): the volume over the triangle is only 1/3.
Pause: let the viewer predict first.
Above the diagonal, all the way up: the strip runs to infinity.
Integrate out y and stop: the strip total at each x is the marginal density.
Six steps. Redraw whenever the order changes.