Answer

(a) This problem is almost exactly like the example. Let x be the length of the slice at distance h from the top of the triangle. The blue and pink triangles are similar:

This means

Multiplying both sides by h,

The area of the slice at depth h from the tip of the triangle is

Since h ranges from 0 at the top of the triangle to 6 at the bottom of the triangle, the area of the triangle is

(b) This integral works out to

This agrees with the area we get using the area formula: