Question

Find the volume of the wedge-shaped region (Figure 1) contained in the cylinder x2+y2=81 and bounded above by the plane z=x and below by the xy-plane.

Answer 1

In cylindrical coordinates, the volume is given with

$\displaystyle\int_{\theta=-\pi/2}^{\theta=\pi/2}\int_{r=0}^{r=9}\int_{z=0}^{z=r\cos\theta}r\,\mathrm dz\,\mathrm dr\,\mathrm d\theta$
$=\displaystyle\int_{\theta=-\pi/2}^{\theta=\pi/2}\int_{r=0}^{r=9}r^2\cos\theta\,\mathrm dr\,\mathrm d\theta$
$=\displaystyle\left(\int_{\theta=-\pi/2}^{\theta=\pi/2}\cos\theta\,\mathrm d\theta\right)\left(\int_{r=0}^{r=9}r^2\,\mathrm dr\right)$
$=486$

Answer 2

Answer:

486 units³

Step-by-step explanation:

Data:

Let the plain be given by the following equation:

$x^{2} + y^{2} = 81$

The bounding plane is  z = x or x = z

The volume is given by cyclical integration using the coordinates:

$\int\limits^\theta = \frac{\pi }{2} _\theta = -\frac{\pi }{2} {rdzdr} \, d\theta$

Integrating gives:

= 486  units³