## total mass of a half cylinder

I need to come up with a formula for the total mass of sawdust that is being stored in a storage shed that is a half cylinder shape with a height of r and length of L. The density of the sawdust at any point is proportional to the distance of that point from the floor. The constant of proportionality is k. I know the volume of the shed is equal to 1/2 pi r^2(L) and mass is equal to density times the area. So I thought that density would be r times k and the formula for mass would be

(k r^3 pi L )/2 the text says it should be (2 k r^3 L)/3