Heat the oven to 350 F. Lightly coat a 9-by-13-inch cake pan with cooking spray. Dust with a bit of flour (about 1 tablespoon) and turn the pan over to remove most of the flour.
In a medium saucepan, add the carrots and water. Bring to a boil and cook for 5 minutes. Drain the water and set aside to cool.
In a large bowl, whisk together the flour, sugar, baking soda and cinnamon.
In another bowl, using an electric mixer on low speed, beat together the oil, applesauce, egg whites and vanilla. Add the flour mixture and beat until well blended. Fold in the cooked carrots, walnuts, coconut and pineapple.
Pour the batter into the prepared pan. Bake until a toothpick inserted into the center of the cake comes out clean, about 1 hour. Let cool in the pan on a wire rack for 10 minutes. Remove the cake from the pan and let cool completely on the cooling rack, about 1 hour.
For the frosting: In a small mixing bowl, add cream cheese and beat slowly with an electric mixer. Add the powdered sugar slowly and continue beating. Add the vanilla and lemon juice. Mix until smooth. Frost the top and sides of the cake.