# How do you find the volume of a solid where #x^2+y^2+z^2=9# is bounded in between the two planes #z+2x=2# and #z+2x=3#?

##### 2 Answers

#### Explanation:

Cumbersome integration problems can be handled easily with the so called Monte Carlo method. https://en.wikipedia.org/wiki/Monte_Carlo_integration

This method works as follows.

1) Choose a box which contains the area/volume to be measured

2) Define the area/volume borders/restrictions

3) Generate inside the box, random values for the coordinates.

a) If for this point the restrictions are obeyed, consider this as a successful one

4)Given de box volume

In this case we have the restrictions defining the sought volume

A python program is attached showing the main details.

The result is

#### Explanation:

The present case can be simplified by a coordinate transformation.

Choosing the transformation

builded using one versor normal to the cutting planes

which are

The new system of coordinates

transform the original equations to

Calculating the revolution volume of

between the limits