Surface Integral Help: Area of Sphere Inside Paraboloid (No Quotation Marks)

[Johnny Blade]

Homework Statement

What is the area of the portion of the sphere $$x^{2}+y^{2}+(z-a)^{2}=a^{2}$$ that is inside the paraboloid $$z=x^{2}+y^{2}$$

Homework Equations

$$\int\int_{S} dS$$

The Attempt at a Solution

I used this

$$\int\int_{S} dS=\int\int_{R}\sqrt{f^{2}_{x}+f^{2}_{y}+1}dx dy$$

And got

$$=\int\int_{R}\frac{a}{\sqrt{a^{2}-x^{2}-y^{2}}}dx dy$$

I know that R is the projection of the surface on the xy plane, but I tried a few different ways to compute the boundaries but it never made sense. Maybe I'm just approaching it the wrong way. Anyone can help me with this?

 

[Gib Z]

Can you please show us what you tried? Did you try solving for the bounds algebraically? Draw a diagram to aid you, and solve the equations simultaneously.