Question

Use a triple integral to determine the volume of the region bounded by z =

p

x

2 + y

2 and z = x

2 + y

2

In 1st octant

Answer 1

To convert from cylindrical to rectangular coordinates, we use the equations

$$x=r\cos \theta, y=r\sin \theta, z=z$$

$$z=\sqrt{x^2+y^2}, z=\sqrt{(r\cos \theta)^2+(r\sin \theta)^2}=r$$

$$z=x^2+y^2, z=(r\cos \theta)^2+(r\sin \theta)^2=r^2$$

$$r=r^2, r_1=0, r_2=1$$

$$V=\displaystyle\int_{0}^{\pi/2}d\theta\displaystyle\int_{0}^{1}rdr\displaystyle\int_{r^2}^{r}dz$$

$$=\displaystyle\int_{0}^{\pi/2}d\theta\displaystyle\int_{0}^{1}(r^2-r^3)dr$$

$$=\dfrac{\pi}{2}(\dfrac{1}{3}-\dfrac{1}{4})=\dfrac{\pi}{24}({units}^2)$$