The only way to find the exact volume is not by integration, but by filling the object (i.e. pepsi bottle) with water and pouring it out into a measuring cylinder. This is similar to the problem, allegedly given by Thomas Edison to a young mathematician, to find out the exact volume of his new invention, the light bulb. After spending days finding a function to represent the curved surface of the bulb and then integrating it to calculate the volume he told Edison the answer. Edison then inverted the bulb and filled it with water and then measured the volume of the water, and replied "Yep, that seems about right." Now that's how an engineer would approach it.