Find the volume of the tetrahedron formed by the planes x = 0, y = 0, z = 0, and x + (y/2) + (z/4) = 1.
Question 5.6.4: Find the volume of the tetrahedron formed by the planes x = ...
The tetrahedron is sketched in Figure 3. We see that z ranges from 0 to the plane x+(y / 2)+(z / 4)=1, or z=4[1-x-(y / 2)]. This last plane intersects the x y-plane in a line whose equation (obtained by setting z=0 ) is given by 0=1 -x-(y / 2) or y=2(1-x), so that y ranges from 0 to 2(1-x) (see Figure 4). Finally, this line intersects the x-axis at the point (1,0,0), so that x ranges from 0 to 1 , and we have
\begin{aligned}V &=\int_{0}^{1} \int_{0}^{2(1-x)} \int_{0}^{4(1-x-y / 2)} d z d y d x=\int_{0}^{1} \int_{0}^{2(1-x)} 4\left(1-x-\frac{y}{2}\right) d y d x \\&=4 \int_{0}^{1}\left\{\left.\left(y(1-x)-\frac{y^{2}}{4}\right)\right|_{0} ^{2(1-x)}\right\} d x=4 \int_{0}^{1}\left[2(1-x)^{2}-(1-x)^{2}\right] d x \\&=-\left.\frac{4}{3}(1-x)^{3}\right|_{0} ^{1}=\frac{4}{3}.\end{aligned}REMARK. It was not necessary to integrate in the order z, then y, then x. We could have written, for example,
0 \leq x \leq 1-\frac{y}{2}-\frac{z}{4}.
The intersection of this plane with the yz-plane is the line 0= 1 – (y/2) – (z/4), or z = 4[1 – (y/2)]. The intersection of this line with the y-axis occurs at the point (0,2,0). Thus
\begin{aligned}V &=\int_{0}^{2} \int_{0}^{4(1-y / 2)} \int_{0}^{1-y / 2-z / 4} d x d z d y \\&=\int_{0}^{2} \int_{0}^{4(1-y / 2)}\left(1-\frac{y}{2}-\frac{z}{4}\right) d z d y=\int_{0}^{2}\left\{\left.\left[\left(1-\frac{y}{2}\right) z-\frac{z^{2}}{8}\right]\right|_{0} ^{4(1-y / 2)}\right\} d y \\&=\int_{0}^{2}\left\{4\left(1-\frac{y}{2}\right)^{2}-\frac{16[1-(y / 2)]^{2}}{8}\right\} d y=2 \int_{0}^{2}\left(1-\frac{y}{2}\right)^{2} d y \\&=-\left.\frac{4}{3}(1-x)^{3}\right|_{0} ^{1}=\frac{4}{3}.\end{aligned}We could also integrate in any of four other orders (x y z, y z x, y x z, and z x y) to obtain the same result.
Related Answered Questions
The region is sketched in Figure 5. Since x...
We have density =\alpha \sqrt{x^{2}+y^{2}+z...
In rectangular coordinates, since
\begin{al...
The solid is sketched in Figure 2. The density is ...
We first note that the solid may be written as [la...
We have already found that \mu=\frac{47}{84...
\begin{aligned}V &=\int_{0}^{1} \int_{x...
\begin{aligned}\mu(S) &=\int_{0}^{1} \i...
\begin{aligned}\iiint_{S} 2 x^{3} y^{2} z d...
\begin{aligned}\iiint_{\pi} x y \cos y z d...