Question

Evaluate the triple integral. 10x dv, where e is bounded by the paraboloid x = 2y2 + 2z2 and the plane x = 2. e

Answer 1

Convert to a rotated set of cylindrical coordinates, taking

$x=t$

$y=r\cos\theta$

$z=r\cos\theta$

As in standard cylindrical coordinates, the volume element is the same

$\mathrm dV=\mathrm dx\,\mathrm dy\,\mathrm dz=r\,\mathrm dr\,\mathrm d\theta\,\mathrm dt$

Then the integral is

$\displaystyle\iiint_E10x\,\mathrm dV=10\int_{\theta=0}^{\theta=2\pi}\int_{r=0}^{r=2}\int_{t=2r^2}^{t=2}rt\,\mathrm dt\,\mathrm dr\,\mathrm d\theta=\frac{1040\pi}3$

Answer 2

This question has based on the mathematical principle called the Cylindrical Coordinate System.

What is the Cylindrical coordinate system?

A cylindrical coordinate system is a three-dimensional coordinate system that Indicates point positions by:

• the distance from a selected axis,
• the direction from the axis relative to a selected reference direction, and
• the distance from a selected reference plane that is at a right angle to the axis.

How is the Evaluate the triple integral above evaluated?

The first step is to the integrals into a rotated set of cylindrical coordinates, such that:

x will become t

y = r cos θ

z = r cos θ

Because the volume factor remains the same in comparison to the coordinates of standard cylinders, we have:

dV = dxdydz = rdr dθ dt

Hence the integral becomes,

$\int\limits$$\int\limit$$\int\limits_E {10x} \, dV$ = 1040x/3

Learn more about the Cylindrical coordinate system at:
https://brainly.com/question/14965899