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Section 4-5 : Triple Integrals

Now that we know how to integrate over a two-dimensional region we need to move on to integrating over a three-dimensional region. We used a double integral to integrate over a two-dimensional region and so it shouldn’t be too surprising that we’ll use a triple integral to integrate over a three dimensional region. The notation for the general triple integrals is,

\[\iiint\limits_{E}{{f\left( {x,y,z} \right)\,dV}}\]

Let’s start simple by integrating over the box,

\[B = \left[ {a,b} \right] \times \left[ {c,d} \right] \times \left[ {r,s} \right]\]

Note that when using this notation we list the \(x\)’s first, the \(y\)’s second and the \(z\)’s third.

The triple integral in this case is,

\[\iiint\limits_{B}{{f\left( {x,y,z} \right)\,dV}} = \int_{{\,r}}^{{\,s}}{{\int_{{\,c}}^{{\,d}}{{\int_{{\,a}}^{{\,b}}{{f\left( {x,y,z} \right)dx}}\,dy}}\,dz}}\]

Note that we integrated with respect to \(x\) first, then \(y\), and finally \(z\) here, but in fact there is no reason to the integrals in this order. There are 6 different possible orders to do the integral in and which order you do the integral in will depend upon the function and the order that you feel will be the easiest. We will get the same answer regardless of the order however.

Let’s do a quick example of this type of triple integral.

Example 1 Evaluate the following integral. \[\iiint\limits_{B}{{8xyz\,dV}} \hspace{0.5in} B = \left[ {2,3} \right] \times \left[ {1,2} \right] \times \left[ {0,1} \right]\]
Show Solution

Just to make the point that order doesn’t matter let’s use a different order from that listed above. We’ll do the integral in the following order.

\[\begin{align*}\iiint\limits_{B}{{8xyz\,dV}} & = \int_{{\,1}}^{{\,2}}{{\int_{{\,2}}^{{\,3}}{{\int_{{\,0}}^{1}{{8xyz\,dz}}\,dx}}\,dy}}\\ & = \int_{{\,1}}^{{\,2}}{{\int_{{\,2}}^{{\,3}}{{\left. {4xy{z^2}} \right|_0^1\,dx}}\,dy}}\\ & = \int_{{\,1}}^{{\,2}}{{\int_{{\,2}}^{{\,3}}{{4xy\,dx}}\,dy}}\\ & = \int_{{\,1}}^{{\,2}}{{\left. {2{x^2}y} \right|_2^3\,dy}}\\ & = \int_{{\,1}}^{{\,2}}{{10y\,dy}} = 15\end{align*}\]

Before moving on to more general regions let’s get a nice geometric interpretation about the triple integral out of the way so we can use it in some of the examples to follow.

Fact

The volume of the three-dimensional region \(E\) is given by the integral,

\[V = \iiint\limits_{E}{{\,dV}}\]

Let’s now move on the more general three-dimensional regions. We have three different possibilities for a general region. Here is a sketch of the first possibility.

This is a sketch of a generic solid that is defined in such a way that the top of the solid can always be given by a function defined as $z=u_{2}\left(x,y\right)$ and that the bottom of the solid can always be given by a function defined as $z=u_{1}\left(x,y\right)$.  The sides of the solid is just a vertical surface that connects the top and bottom of the surface.  For this this particular graph the top and bottom are shown as basically flat surfaces but they do not need to be flat.  Below the solid is a general region in the xy-plane denoted as D.  This is the region that the points (x,y) come from to sketch the top/bottom surface of the solid.

In this case we define the region \(E\) as follows,

\[E = \left\{ {\left( {x,y,z} \right)|\left( {x,y} \right) \in D,\,\,\,{u_1}\left( {x,y} \right) \le z \le {u_2}\left( {x,y} \right)} \right\}\]

where \(\left( {x,y} \right) \in D\) is the notation that means that the point \(\left( {x,y} \right)\) lies in the region \(D\) from the \(xy\)-plane. In this case we will evaluate the triple integral as follows,

\[\iiint\limits_{E}{{f\left( {x,y,z} \right)\,dV}} = \iint\limits_{D}{{\left[ {\int_{{\,{u_1}\left( {x,y} \right)}}^{{\,{u_2}\left( {x,y} \right)}}{{f\left( {x,y,z} \right)\,dz}}} \right]\,dA}}\]

where the double integral can be evaluated in any of the methods that we saw in the previous couple of sections. In other words, we can integrate first with respect to \(x\), we can integrate first with respect to \(y\), or we can use polar coordinates as needed.

Example 2 Evaluate \(\displaystyle \iiint\limits_{E}{{2x\,dV}}\) where \(E\) is the region under the plane \(2x + 3y + z = 6\) that lies in the first octant.
Show Solution

We should first define octant. Just as the two-dimensional coordinates system can be divided into four quadrants the three-dimensional coordinate system can be divided into eight octants. The first octant is the octant in which all three of the coordinates are positive.

Here is a sketch of the plane in the first octant.

This is a graph with the standard 3D coordinate system.  The positive z-axis is straight up, the positive x-axis moves downwards and slightly to the right and positive y-axis moves off nearly horizontal to the right.This is a sketch of the portion of the plane that is in the 1st octant.  It is a triangle with vertices at (0,0,6), (3,0,0) and (0,2,0). This is a boxed 3D coordinate system.  The z-axis is right vertical edge of the box, the x-axis is the top right edge of the box and the y-axis is the bottom front edge of the box.  This is a sketch of the portion of the plane that is in the 1st octant.  It is a triangle with vertices at (0,0,6), (3,0,0) and (0,2,0).

We now need to determine the region \(D\) in the \(xy\)-plane. We can get a visualization of the region by pretending to look straight down on the object from above. What we see will be the region \(D\) in the \(xy\)-plane. So \(D\) will be the triangle with vertices at \(\left( {0,0} \right)\), \(\left( {3,0} \right)\), and \(\left( {0,2} \right)\). Here is a sketch of \(D\).

This is the 2D graph on the domain 0<x<3 of a triangle with vertices (0,0), (3,0) and (0,2).  The top edge of the triangle has equation $y=-\frac{2}{3}x+2$ or $x=-\frac{3}{2}x+3$.  The bottom edge is the x-axis and the left edge is the y-axis.  The triangle has been shaded in.

Now we need the limits of integration. Since we are under the plane and in the first octant (so we’re above the plane \(z = 0\)) we have the following limits for \(z\).

\[0 \le z \le 6 - 2x - 3y\]

We can integrate the double integral over \(D\) using either of the following two sets of inequalities.

\[\begin{matrix} \begin{aligned} & \,\,\,\,\,\,0\le x\le 3 \\ & 0\le y\le -\frac{2}{3}x+2 \\ \end{aligned} & \hspace{0.5in} & \begin{aligned} & 0\le x\le -\frac{3}{2}y+3 \\ & \,\,\,\,\,\,0\le y\le 2 \\ \end{aligned} \\ \end{matrix}\]

Since neither really holds an advantage over the other we’ll use the first one. The integral is then,

\[\begin{align*}\iiint\limits_{E}{{2x\,dV}} &= \iint\limits_{D}{{\left[ {\int_{{\,0}}^{{6 - 2x - 3y}}{{2x\,dz}}} \right]\,dA}}\\ & = \iint\limits_{D}{{\left. {2xz} \right|_0^{6 - 2x - 3y}\,dA}}\\ & = \int_{{\,0}}^{{\,3}}{{\int_{0}^{{ - \frac{2}{3}x + 2}}{{2x\left( {6 - 2x - 3y} \right)\,dy}}\,dx}}\\ & = \int_{0}^{3}{{\left. {\left( {12xy - 4{x^2}y - 3x{y^2}} \right)} \right|_0^{ - \frac{2}{3}x + 2}\,dx}}\\ & = \int_{0}^{3}{{\frac{4}{3}{x^3} - 8{x^2} + 12x\,dx}}\\ & = \left. {\left( {\frac{1}{3}{x^4} - \frac{8}{3}{x^3} + 6{x^2}} \right)} \right|_0^3\\ & = 9\end{align*}\]

Let’s now move onto the second possible three-dimensional region we may run into for triple integrals. Here is a sketch of this region.

This is a graph with the standard 3D coordinate system.  The positive z-axis is straight up, the positive x-axis moves off to the left and slightly downward and positive y-axis moves off the right and slightly downward.  This is a sketch of a generic solid that is defined in such a way that the left/front face of the solid can always be given by a function defined as $x=u_{2}\left(y,z\right)$ and that the right/back face of the solid can always be given by a function defined as $x=u_{1}\left(y,z\right)$.  The sides of the solid is just a horizontal surface that connects the left/front and right/back face of the surface.  For this this particular graph the left and right face are shown as basically flat surfaces but they do not need to be flat.  Behind the solid is a general region in the yz-plane denoted as D.  This is the region that the points (y,z) come from to sketch the left/right faces of the solid.

For this possibility we define the region \(E\) as follows,

\[E = \left\{ {\left( {x,y,z} \right)|\left( {y,z} \right) \in D,\,\,\,{u_1}\left( {y,z} \right) \le x \le {u_2}\left( {y,z} \right)} \right\}\]

So, the region \(D\) will be a region in the \(yz\)-plane. Here is how we will evaluate these integrals.

\[\iiint\limits_{E}{{f\left( {x,y,z} \right)\,dV}} = \iint\limits_{D}{{\left[ {\int_{{\,{u_1}\left( {y,z} \right)}}^{{\,{u_2}\left( {y,z} \right)}}{{f\left( {x,y,z} \right)\,dx}}} \right]\,dA}}\]

As with the first possibility we will have two options for doing the double integral in the \(yz\)-plane as well as the option of using polar coordinates if needed.

Example 3 Determine the volume of the region that lies behind the plane \(x + y + z = 8\) and in front of the region in the \(yz\)-plane that is bounded by \(\displaystyle z = \frac{3}{2}\,\,\sqrt y \) and \(\displaystyle z = \frac{3}{4}y\).
Show Solution

In this case we’ve been given \(D\) and so we won’t have to really work to find that. Here is a sketch of the region \(D\) as well as a quick sketch of the plane and the curves defining \(D\) projected out past the plane so we can get an idea of what the region we’re dealing with looks like.

This is the 2D graph on a yz-axis system with domain 0<y<4 of $z=\frac{3}{2} \sqrt{y}$ and $z=\frac{3}{4}y$.  In this domain the graph of $z=\frac{3}{2} \sqrt{y}$ is always larger than the graph of $z=\frac{3}{4}y$.  The area between the two functions has been shaded in. This is a graph with the standard 3D coordinate system.  The positive z-axis is straight up, the positive x-axis moves downwards and slightly to the right and positive y-axis moves off nearly horizontal to the right.  This is a sketch of the portion of the plane from the problem statement that is in the 1st octant.  It is a triangle with vertices at (0,0,8), (8,0,0) and (0,8,0).  Also included in the graph is a solid whose cross section is given in the 2D graph of the region D.  The solid starts in the yz-plane behind the plane given in the problem statement and moves in the positive x direction.  Eventually it starts intersecting the plane and stops moving when it has completely come out from behind the plane.  The solid looks somewhat like an airplane wing with on edge on the x-axis and other other angles upwards.

Now, the graph of the region above is all okay, but it doesn’t really show us what the region is. So, here is a sketch of the region itself.

This is a graph with the standard 3D coordinate system.  The positive z-axis is straight up, the positive x-axis moves downwards and slightly to the right and positive y-axis moves off nearly horizontal to the right. This is a sketch of the solid described in the 3D graph above.  However, this time the plane from the problem statement is only graph where the solid actually intersects it.  In other words, the plane forms the front face of the solid.  Again, the solid is a shaped somewhat like an airplane wing with one edge on the x-axis over the range 0<x<8.  The solid then angles upwards and as it does the front fact recedes until it reaches approximately the point (2,4,3).  From this point the full solid goes back to the yz-plane. This is a boxed 3D coordinate system.  The z-axis is right vertical edge of the box, the x-axis is the top right edge of the box and the y-axis is the bottom front edge of the box.This is a sketch of the solid described in the 3D graph above.  However, this time the plane from the problem statement is only graph where the solid actually intersects it.  In other words, the plane forms the front face of the solid.  Again, the solid is a shaped somewhat like an airplane wing with one edge on the x-axis over the range 0<x<8.  The solid then angles upwards and as it does the front fact recedes until it reaches approximately the point (2,4,3).  From this point the full solid goes back to the yz-plane.

Here are the limits for each of the variables.

\[\begin{array}{c}0 \le y \le 4\\ \displaystyle \frac{3}{4}y \le z \le \frac{3}{2}\sqrt y \\ 0 \le x \le 8 - y - z\end{array}\]

The volume is then,

\[\begin{align*}V &= \iiint\limits_{E}{{\,dV}} = \iint\limits_{D}{{\left[ {\int_{0}^{{8 - y - z}}{{\,dx}}} \right]\,dA}}\\ & = \int_{0}^{4}{{\int_{{{{3y}}/{4}\;}}^{{{{3\sqrt y }}/{2}\;}}{{8 - y - z\,dz}}\,dy}}\\ & = \int_{0}^{4}{{\left. {\left( {8z - yz - \frac{1}{2}{z^2}} \right)} \right|_{\frac{{3y}}{4}}^{\frac{{3\sqrt y }}{2}}\,dy}}\\ & = \int_{0}^{4}{{12{y^{\frac{1}{2}}} - \frac{{57}}{8}y - \frac{3}{2}{y^{\frac{3}{2}}} + \frac{{33}}{{32}}{y^2}\,dy}}\\ & = \left. {\left( {8{y^{\frac{3}{2}}} - \frac{{57}}{{16}}{y^2} - \frac{3}{5}{y^{\frac{5}{2}}} + \frac{{11}}{{32}}{y^3}} \right)} \right|_0^4 = \frac{{49}}{5}\end{align*}\]

We now need to look at the third (and final) possible three-dimensional region we may run into for triple integrals. Here is a sketch of this region.

This is a graph with the standard 3D coordinate system.  The positive z-axis is straight up, the positive x-axis moves off to the left and slightly downward and positive y-axis moves off the right and slightly downward.  This is a sketch of a generic solid that is defined in such a way that the right/front face of the solid can always be given by a function defined as $y=u_{2}\left(x,z\right)$ and that the left/back face of the solid can always be given by a function defined as $y=u_{1}\left(x,z\right)$.  The sides of the solid is just a horizontal surface that connects the right/front and left/back face of the surface.  For this this particular graph the left and right face are shown as basically flat surfaces but they do not need to be flat.  Behind the solid is a general region in the xz-plane denoted as D.  This is the region that the points (x,z) come from to sketch the left/right faces of the solid.

In this final case \(E\) is defined as,

\[E = \left\{ {\left( {x,y,z} \right)|\left( {x,z} \right) \in D,\,\,\,{u_1}\left( {x,z} \right) \le y \le {u_2}\left( {x,z} \right)} \right\}\]

and here the region \(D\) will be a region in the \(xz\)-plane. Here is how we will evaluate these integrals.

\[\iiint\limits_{E}{{f\left( {x,y,z} \right)\,dV}} = \iint\limits_{D}{{\left[ {\int_{{\,{u_1}\left( {x,z} \right)}}^{{\,{u_2}\left( {x,z} \right)}}{{f\left( {x,y,z} \right)\,dy}}} \right]\,dA}}\]

where we will can use either of the two possible orders for integrating \(D\) in the \(xz\)-plane or we can use polar coordinates if needed.

Example 4 Evaluate \(\displaystyle \iiint\limits_{E}{{\sqrt {3{x^2} + 3{z^2}} \,dV}}\) where \(E\) is the solid bounded by \(y = 2{x^2} + 2{z^2}\) and the plane \(y = 8\).
Show Solution

Here is a sketch of the solid \(E\).

This is a graph with the standard 3D coordinate system.  The positive z-axis is straight up, the positive x-axis moves off to the left and slightly downward and positive y-axis moves off the right and slightly downward.  This is the sketch of the vaguely cup shaped solid from the elliptic paraboloid given in the problem statement.  It starts at the origin, is centered on the y-axis and opens up in the positive y direction.  The solid is capped at y=8 by a disk. This is a boxed 3D coordinate system.  The z-axis is left vertical edge of the box, the x-axis is the bottom right edge of the box and the y-axis is the top front edge of the box.  This is the sketch of the vaguely cup shaped solid from the elliptic paraboloid given in the problem statement.  It starts at the origin, is centered on the y-axis and opens up in the positive y direction.  The solid is capped at y=8 by a disk.

The region \(D\) in the \(xz\)-plane can be found by “standing” in front of this solid and we can see that \(D\) will be a disk in the \(xz\)-plane. This disk will come from the front of the solid and we can determine the equation of the disk by setting the elliptic paraboloid and the plane equal.

\[2{x^2} + 2{z^2} = 8\hspace{0.25in} \Rightarrow \hspace{0.25in}{x^2} + {z^2} = 4\]

This region, as well as the integrand, both seems to suggest that we should use something like polar coordinates. However, we are in the \(xz\)-plane and we’ve only seen polar coordinates in the \(xy\)-plane. This is not a problem. We can always “translate” them over to the \(xz\)-plane with the following definition.

\[x = r\cos \theta \hspace{0.25in}\hspace{0.25in}z = r\sin \theta \]

Since the region doesn’t have \(y\)’s we will let \(z\) take the place of \(y\) in all the formulas. Note that these definitions also lead to the formula,

\[{x^2} + {z^2} = {r^2}\]

With this in hand we can arrive at the limits of the variables that we’ll need for this integral.

\[\begin{array}{c}2{x^2} + 2{z^2} \le y \le 8\\ 0 \le r \le 2\\ 0 \le \theta \le 2\pi \end{array}\]

The integral is then,

\[\begin{align*}\iiint\limits_{E}{{\sqrt {3{x^2} + 3{z^2}} \,dV}} & = \iint\limits_{D}{{\left[ {\int_{{2{x^2} + 2{z^2}}}^{{\,8}}{{\sqrt {3{x^2} + 3{z^2}} \,dy}}} \right]\,dA}}\\ & = \iint\limits_{D}{{\left. {\left( {y\sqrt {3{x^2} + 3{z^2}} } \right)} \right|_{2{x^2} + 2{z^2}}^8\,dA}}\\ & = \iint\limits_{D}{{\sqrt {3\left( {{x^2} + {z^2}} \right)} \left( {8 - \left( {2{x^2} + 2{z^2}} \right)} \right)\,dA}}\end{align*}\]

Now, since we are going to do the double integral in polar coordinates let’s get everything converted over to polar coordinates. The integrand is,

\[\begin{align*}\sqrt {3\left( {{x^2} + {z^2}} \right)} \left( {8 - \left( {2{x^2} + 2{z^2}} \right)} \right) &= \sqrt {3{r^2}} \left( {8 - 2{r^2}} \right)\\ & = \sqrt 3 \,\,r\left( {8 - 2{r^2}} \right)\\ & = \sqrt 3 \left( {8r - 2{r^3}} \right)\end{align*}\]

The integral is then,

\[\begin{align*}\iiint\limits_{E}{{\sqrt {3{x^2} + 3{z^2}} \,dV}} & = \iint\limits_{D}{{\sqrt 3 \,\,\left( {8r - 2{r^3}} \right)dA}}\\ & = \sqrt 3 \int_{{\,0}}^{{\,2\pi }}{{\int_{{\,0}}^{{\,2}}{{\left( {8r - 2{r^3}} \right)r\,dr}}\,d\theta }}\\ & = \sqrt 3 \int_{{\,0}}^{{\,2\pi }}{{\left. {\left( {\frac{8}{3}{r^3} - \frac{2}{5}{r^5}} \right)} \right|_0^2\,d\theta }}\\ & = \sqrt 3 \int_{{\,0}}^{{\,2\pi }}{{\frac{{128}}{{15}}\,d\theta }}\\ & = \frac{{256\sqrt 3 \,\pi }}{{15}}\end{align*}\]