= Solution
The inequalities describing the region are $0\leq y\leq1$ and $2y\leq x\leq2$. It is the closed <triangle> bounded by $y=0$, $x=2$ and $x=2y$, with vertices
$$
\boxed{(0,0),\qquad(2,0),\qquad(2,1).}
$$
A horizontal slice runs from the sloping edge $x=2y$ to the vertical edge $x=2$.
\Image[/past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-1-q12-region-and-cup.png]
{title=The triangular integration region and the cup cross-section, including its solid base, in Question 12}
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