Volumes by Slicing Diagram

remember that a and h are constants. The area of the purple square that is 2y units per side is A = 4y². We will integrate that from 0 to h but we need it in terms of x, a, and h not in terms of y.

Comparing the dimensions of the purple square and the base square gives us:

Now we can write A(y) = 4y² in terms of x using the substitution for y. We get

Now we integrate this from x = 0 to x = h and we'll have the volume of the pyramid. Recall that a and h are constants -- values for the dimensions of the pyramid.

Had we been told that the base of the pyramid is a square 9 meters per side and height of the pyramid is 10 meters, we would substitute a = 9 and h = 10 to find the volume.