Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2012 based on cross section of solids and volumes.
Cube ABCDEFGH labeled as shown below has edge length 1 and is cut by a plane passing through vertex D and the midpoints M and N of AB and CG respectively. The plane divides the cube into solids. The volume of the larger of the two solids can be written in the form \(\frac{p}{q}\) where p and q are relatively prime. find p+q.
Calculus
Algebra
Geometry
Answer: 89.
AIME, 2012, Question 8
Calculus Vol 1 and 2 by Apostle
DMN plane cuts the section of solid with \(z=\frac{y}{2}-\frac{x}{4}\) intersects base at \(y=\frac{x}{2}\)
\(V=\int_0^1\int_{\frac{x}{2}}^1\int_0^{\frac{y}{2}-\frac{x}{4}}{d}x{d}y{d}z\)=\(\frac{7}{48}\)
other portion 1-\(\frac{7}{48}\)=\(\frac{41}{48}\) then 41+48=89.
