$$\iiint_E x^2dV, \text{where E is the solid tetrahedron with vertices }(0,0,0), (1,0,0), (0,1,0), \text{and (0,0,1)}$$
I need some assistance on setting up the limits. If someone could help me learn how to set up my limits of integration, that would be great. I should be able to integrate it just fine.
[Math] Evaluate the triple integral, tetrahedron
integrationmultivariable-calculus
Related Solutions
Here is your region:
Notice, it is bounded by the lines $y=0$ and $y=1$, so those are our bounds for $y$. Next, for any specific $y_0$, we are considering the $x$ values that range from $0$ to $y_0$, so our bounds for $x$ are $0<x<y$. Is that clear? The integral becomes:
$$\int_{0}^1\int_0^ye^{\frac{x}{y}}dxdy$$
To get an idea what to do in 3D, try to understand the 2D case first and try to generalize. When integrating over the triangle with vertices $(0,0)$, $(0,1)$ and $(1,0)$, it is often a good idea to first let $x$ go from zero to $1-y$ and then let $y$ go from zero to 1.
In your case, you can proceed analogously: let $x$ range in $[0,1-y-z]$, then $y$ in $[0,1-z]$ and finally $z$ in $[0,1]$. That is, $$ \int_{\text{tetrahedron}}f(x,y,z)dxdydz = \int_0^1\int_0^{1-z}\int_0^{1-y-z}f(x,y,z)dxdydz. $$ In your case $f(x,y,z)=xyz$. To make the iterated integral structure clear, you can write it as $$ \int_0^1\left(\int_0^{1-z}\left(\int_0^{1-y-z}xyzdx\right)dy\right)dz $$ and start by doing the innermost integral. The first integral is $\int_0^{1-y-z}xyzdx=\frac12yz(1-y-z)^2$. If you get this right, you are on the right track.
Best Answer
Your tetrahedron is bordered by the coordinate planes, and the plane $x+y+z=1$. Sketch a picture! So $x\ge0,$ $y\ge0$, $z\ge0$ and $x+y+z\le1$. [Edit: If you have trouble with this step, then go and review how you determine the equation of a plane given that you know three points on it. Here the coordinate planes $x=0$, $y=0$ and $z=0$ stand out. The fourth plane passes via the points $(1,0,0), (0,1,0)$ and $(0,0,1)$. If everything else fails, you can go through that process. Here with a bit of experience you should notice that the coordinates of these three points sum up to $1$. That gives you the equation of the last plane./Edit]
As a next step you should ask yourself the questions:
This will give you the limits, if you first integrate w.r.t. $z$, then $y$, last $x$. If you prefer (sometimes it will be to your advantage), you can process the variables in a different order.