Using double integrals to calculate the volume of a solid over general regions

 In a previous post we define the notion of general regions that allows us to calculate doubles integrals over these types of regions. This enables also to calculate the volume of a solid over these regions. Let's dive right into the solution of a problem.

Example

Solution

The solid is limited by the plane x = 0 or yz. the plane  y = 0 or xz, the plane z = 0 or xy and the plane 2x +3y+z = 6. First  we start by drawing the axes of coordinates then we draw part of the plane 2x+3y+z = 0. In order to do the latter, let's determine the coordinates intercepts.

Let's connect the points to form a triangle:

The triangle represents part of the plane 2x+3y+z = 6.. The plane intercepts the planes x = 0, y = 0, z = 0. This gives us the other faces of the tetrahedron, The base is limited by the x axis, the y axis and the line 2x + 3y = 6 as seen on the right. 

Looking at the figure we see a solid with 4 triangular faces, It's called a tetrahedron or triangular pyramid. It consists of the three coordinates planes, the plane z = -2x-3y+6 with the bases limited by the lines y =o, x = 0 and the line 2x +3y = 6.

Practice

Changing the order of integration in the calculation of a double integral

 As we have seen before, sometimes one order of integration can make the calculation of a double integral easier than another order. Another time the order of integration doesn't matter. It iis important to recognize when a change in the order of integration can facilitate the work.

Example

Solution

Decomposing regions into smaller regions in the calculation of double integrals

 Decomposition of regions into smaller regions in the calculation of double integrals

Goal: Decompose a region into smaller regions in the calculation of double integrals.

The properties of double integrals over rectangular regions are valid for the the calculation of double integrals over non rectangular bounded region on a plane. The property 3 of double integrals over rectangular region states:

The following theorem establishes the same property for double integrals over non rectangular regions on a plane.

Theorem 

Suppose the region D can be expressed as D = D₁∪D₂ where D₁ and D₂ do not overlap except at their boundaries.

Example

Solution

Here D₁ is type 1 and D₂ and D₃ are type 2. Hence:

Evaluating a double integral over a type II region

 We continue in the evaluation of double integrals over non-rectangular regions. In the previous  post, we solve a double integral over a type I region. In this post, we evaluate the double integral of a function over a type II region.

Example

Solution

Practice

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