Find the volume of the solid bounded by the planes x0y0z0x0y0z0 and x y z7


Introduction

In this lesson, we will be finding the volume of the solid that is bounded by the planes x0y0z0x0y0z0 and x y z7. We will be using the method of slicing to find the volume of this solid.

The Volume of the solid


We need to find the volume of the solid that is bounded by the planes x0y0z0, x0y0z7, x0y7z0 and x7yz0.

We can do this by integrating over each of the three variables:

int int int (x*y*z) dydzdx

The dimensions of the solid

We want to find the volume of the solid that is bounded by the planes x0y0z0, x0y0z7, x0yz7, xyz7. To do this, we need to find the dimensions of the solid first. We can see that the planes x0y0z0 and xyz7 intersect at the point (0, 0, 7), so we know that one dimension of the solid is 7. The other two dimensions can be found by finding the intersection of the planes x0y0z7 and xyz7. This gives us the point (1/3, 1/3, 7), so we know that the other two dimensions are 1/3. Thus, the volume of the solid is 71/31/3 = 1.

The calculation of the volume

To calculate the volume of the solid, we will use the formula for the volume of a rectangular solid. In this case, the sides of the rectangular solid are x=0,y=0,z=0,x=0,y=0,z=7. The volume of the rectangular solid is calculated using the formula V=xyz. We plug in the known values to get: V=(0)(0)(7)=0

The conclusion


To find the volume of the solid, we need to find the limits of integration. In this case, the bounds are x=0, y=0, z=0 and x=7, y=7, z=7.

We can use these bounds to set up the integral:

$\int\int\int_D f(x,y,z) \,dx\,dy\,dz$

where D is the region bounded by the planes.


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