Help with Limits of Integration

[viciousp]

Homework Statement

A Hemispheric bowl has a radius of a and a depth of h. Find the Volume

Homework Equations

$$r= \sqrt{a^2-y^2}$$

$$\pi \int{(\sqrt{a^2-y^2)^2)}$$

The Attempt at a Solution

I solved the integral using the limits h and 0 and got $$\frac{\pi*h(3a^2-h^2)}{3}$$.

But the book used h-a and -a and got $$\frac{\pi*h^2(3a-h)}{3}$$

My main question is if my limits are acceptable and if the book has the correct one why must I use their limits.

 

[dynamicsolo]

If you think about a sphere of radius a, you found the volume of the spherical cap from the "north pole" at y = a down to y = a-h, while the solver for the textbook chose the "south pole" cap from y = -a up to y = h-a .

You call your levels y' = 0 and y' = h , which means you are just choosing a different reference level for "zero" and have the direction of y reversed from the textbook's solver. So fundamentally, there is no difference between your approach and theirs. The transformation suggested by the values above to go from their system to yours is y' = y + h . Your choice of coordinates is completely legitimate.

(Found the problem!)
EDIT: However, while the expression $$r = \sqrt{a^2 - y^2}$$ works for the solver's choice of coordinates, which measures from the center of the sphere, for your choice, measurements are made from a pole of the sphere. So you must modify your expression to

$$r = \sqrt{a^2 - (a-y)^2} .$$

 

[viciousp]

Thanks