Volume of a paraboloid and circle

Question

I need to calculate the volume of a paraboloid

$$z \geq \frac{x^2}{a^2} + \frac{y^2}{a^2}$$

bounded by a circle

$$x^2+y^2 \leq R^2$$

where $R$ is the circle radius.

I think that use a polar coordinates, but I do not how make the integrals.

Answer 1

The expression "bounded by a circle" is not correct. I assume, what you mean using this is expression, is that your region is bounded by the paraboloid $ z = \cfrac{x^2}{a^2} + \cfrac{y^2}{a^2} $ and the plane $ z = \cfrac{R^2}{a^2} $, so that the condition $ x^2+y^2 \leq R^2 $ is satisfied.

If so, the region can be described as follows: $$ D = \Bigg \{ (x,y,z) \in \mathbb{R^3} : \cfrac{x^2}{a^2} + \cfrac{y^2}{a^2} \leq z \leq \cfrac{R^2}{a^2} \Bigg \} $$

You can calculate volume using either polar or cylindrical coordinates.

Using polar coordinates:

$$ V = \iiint \limits_D \; \mathrm{d}x \mathrm{d}y \mathrm{d}z = \int_{-R}^R \int_{-\sqrt {R^2-x^2}}^{\sqrt {R^2-x^2}} \int_{\frac{x^2}{a^2} + \frac{y^2}{a^2}}^\frac{R^2}{a^2} \; \mathrm{d}z \mathrm{d}y \mathrm{d}x = \iint \limits_{x^2+y^2 \leq R^2} \cfrac{R^2-x^2-y^2}{a^2} \mathrm{d}y \mathrm{d}x $$ We have $ x=r\cosθ , y=r\sin θ $ and $ \cfrac{\partial (x,y)}{\partial (r,θ)}=r $.

For the domain of integration $T$ we have: $$ T = \Bigg \{ (x,y) \in \mathbb{R^2} : x^2+y^2 \leq R^2 \Bigg \} = \Bigg \{ (r,θ) \in \mathbb{R^2} : 0 \leq θ \leq 2π, 0 \leq r \leq R \Bigg \} $$ Therefore, $$ V = \displaystyle \int_{0}^{2π} \int_{0}^{R} \cfrac {R^2-r^2}{a^2} r \; \mathrm{d}r \mathrm{d}θ = \, ... = \cfrac{πR^4}{2a^2} $$ Using cylindrical coordinates:

We have $ x=r\cosθ , y=r\sin θ, z = z $ and $ \cfrac{\partial (x,y,z)}{\partial (r,θ,z)}=r $. So, the region can now be given by: $$ D = \Bigg \{ (r,θ,z) \in \mathbb{R^3} : 0 \leq θ \leq 2π, 0 \leq z \leq \cfrac{R^2}{a^2}, 0 \leq r \leq r(z)= a\sqrt z \Bigg \} $$
Therefore, $$ V = \displaystyle \int_{0}^{2π} \int_{0}^{\frac{R^2}{a^2}} \int_{0}^{a\sqrt z} r \; \mathrm{d}r \mathrm{d}z \mathrm{d}θ = \, ... = \cfrac{πR^4}{2a^2} $$

So, the requested volume equals $\cfrac{πR^4}{2a^2}$.