Triple Integrals

Set up and evaluate an iterated integral that will evaluate the triple integral of $f(x,y,z) = x-y+2z$ over the box $B = [-2,3] \times [1,4] \times [0,2]$.

Consider the solid that lies between the paraboloids $z = g(x,y) = x^2 + y^2$ and $z = f(x,y) = 8 - 3x^2 - 3y^2$.

1. By eliminating the variable $z$, determine the curve of intersection between the two paraboloids, and sketch this curve in the $xy$-plane.

2. Set up, but do not evaluate, an iterated integral expression whose value determines the mass of the solid, integrating first with respect to $z$, then $y$, then $x$. Assume the the solid's density is given by $\delta(x,y,z) = \frac{1}{x^2 + y^2 + z^2 + 1}$.

3. S…

In each of the following problems, your task is to

1. Sketch, by hand, the region over which you integrate

2. set up iterated integral expressions which, when evaluated, will determine the value sought

3. use appropriate technology to evaluate each iterated integral expression you develop

In some problems you may be able to use a double rather than a triple integral, and polar coordinates may be helpful in som…