Finding the volume of a R 3 triangleI have a triangle ABC defined with the points A = ( 2.4 , − 5.4 , 6 ), B = ( 0 , 1.1 , 3.2 ), C = ( − 7.6 , 3 , 0 ).And I'm asked to find the volume of the solid of R 3 given by the points between the plane that is made by the points of the triangle and the plaze z = 0 (The same points of the triangle with all the z = 0)I think I should use integrals but I don't know from which regions to do the triple integration.

Question

Finding the volume of a             R        3   triangleI have a triangle ABC defined with the points   A  =  (  2.4  ,  −  5.4  ,  6  ),   B  =  (  0  ,  1.1  ,  3.2  ),   C  =  (  −  7.6  ,  3  ,  0  ).And I&#039;m asked to find the volume of the solid of             R        3   given by the points between the plane that is made by the points of the triangle and the plaze       z    =    0   (The same points of the triangle with all the   z  =  0)I think I should use integrals but I don&#039;t know from which regions to do the triple integration.

wietselau · Accepted Answer

Step 1The area of a triangle with vertices A, B, C in             R        3   can be obtained using the cross product:      S    u    r    f    a    c    e    =      1    2    ‖            A      B        →    ×            A      C        →    ‖  .Also, the volume of a tetrahedron defined by four points A, B, C, D, can be computed using       V    o    l    u    m    e    =      1    6        |    det  (            A      B        →    ,            A      C        →    ,            A      D        →    )      |    .

Taliyah Reyes · Accepted Answer

Step 1First, write down the inequalities that define the triangle. This is the region you&#039;ll be integrating across.Step 2Second, compute the function that tells you how far a point, (x, y, z) is from the plane   z  =  0. This will be the function you integrate.Step 3Third, put these two together and try computing the integral.

Given a triangle ABC defined with the points A=(2.4, -5.4, 6), B=(0, 1.1, 3.2), C=(-7.6, 3,0). Find the volume of the solid of mathbb{R}^3 given by the points between the plane that is made by the points of the triangle and the plaze z=0 (The same points of the triangle with all the z=0)

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