Triple Integrals : Finding Volume of Solids with Boundaries
1) Evaluate the triple integral e^(1-(x^2)-(y^2)) dxdydz with T the solid enclosed by z=0 and z= 4-(x^2)-(y^2)
2) Find the volume of the solid bounded above and below by the cone (z^2) = (x^2) + (y^2), and the side by y=0 and y= square root(4-(x^2)-(z^2))
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a) We want to solve:
Well, the form x^2+y^2 leads us to this fact that the change of variables as x=r*cos(Ө), y=r*sin(Ө) and z=z could be useful. Because there have been no words on the limits of x and y, we consider 0≤Ө≤2п and 0≤r<∞ to cover the entire space. Then we naturally have: 0≤z≤(4-r^2)
We also have to consider that the jacobian of this change of variables is r. ...
Solution Summary
A triple integral is evaluated, and the volume of a solid is obtained.