I am struggling a lot with triple integrals. I can evaluate them, but I find it extremely difficult to write the triple integral. I cannot visualize them.
An example question: $\iiint_E6xy\;dV$, where $E$ lies under the plane $z = 1 + x + y$ and above the region in the xy-plane bounded by the curves $y=\sqrt{x}$, $y=0$, $x=1$.
Since the solid lies under the plane, the upper limit for $z$ would be $1 + x + y$. Since the lower bounded region is in the xy plane, the lower bound would be 0? Thus, $\iiint_0^{1+x+y}6xy\;dz...$
For the other integrals x and y, I would simply sketch the xy region and integrate the same as double integrals. Which I get: $\int_0^1\int_{\sqrt{x}}^1\int_0^{1+x+y}6xy\;dz\;dy\;dx$
Are these limits of integration correct? Is the approach correct? The professor taught by visuals, which I find extremely difficult to do in an xyz plane.