Question

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

z

=

√

x

2

+

y

2

and

z

=

x

2

+

y

2

in 1st octant.

Answer 1

By equating the coordinates $z,$ we get the following equation:

$$\sqrt{x^2+y^2}=x^2+y^2$$

$$x^2+y^2=0\ or\ x^2+y^2=1$$

Using the cylindrical coordinates

$$V=\int\int\int_EdV=$$

$$=\displaystyle\int_{0}^{\pi/2}\displaystyle\int_{0}^{1}\displaystyle\int_{r^2}^{r}rdzdrd\theta$$

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

$$=\displaystyle\int_{0}^{\pi/2}\bigg[\dfrac{r^3}{3}-\dfrac{r^4}{4}\bigg]\begin{matrix} 1 \\ 0 \end{matrix}d\theta$$

$$=\dfrac{1}{12}\big[\theta\big]\begin{matrix} \pi/2 \\ 0 \end{matrix}=\dfrac{\pi}{24}(units^3)$$

$V=\dfrac{\pi}{24}$ cubic units