4.7 Triple Integrals

Triple Integrals

Introduction

Triple Riemann Sums and Triple Integrals

In this activity, we want to try to estimate the mass of large piece of granite. Granite is composed of different minerals such as feldspar and quartz that give distinctive patterns of color and texture. The large piece of granite we are looking at is 4 feet wide, six feet deep, and 8 feet tall which we will describe by

. This very special piece of granite formed in a region with many geological folds and has its density given by $\delta(x,y,z)= 163+3\sin(xy)+\frac{2z}{3}$. The units for $x,y,z$ are measured in feet and the density is given in pounds per cubic foot.

For a solid of constant density, we can find the mass by multiplying the density and volume. For our block of granite, the density varies from point to point based on the function $\delta(x,y,z)$. In this activity, we will approximate the mass of the block (step 1 of the Proposition 2.1.1) by slicing the solid into smaller pieces on which there are smaller density changes and thus the density is closer to constant.

For a first approximation using smaller pieces, we partition the block as follows - the width, given by the $x$-interval $[0,4]$, into two subintervals of equal length - the depth, given by the $y$-interval $[0,6]$ into three subintervals of equal length - the height, given by the $z$-interval $[0,8]$ into two subintervals of equal length This partitions the box $B$ into sub-boxes as shown in Figure.

Let $0=x_0 \lt x_1 \lt x_2=4$ be the endpoints of the $x$-subintervals of $[0,4]$ after partitioning. Label these endpoints on Figure. Repeat this process with $0=y_0 \lt y_1 \lt y_2 \lt y_3=6$ and $0=z_0 \lt z_1 \lt z_2=8$.

In this example, we will find the mass of the tetrahedron $S$ in the first octant bounded by the coordinate planes and the plane $x + 2 y + 3 z = 6$ if the density at point $(x,y,z)$ is given by $\delta(x, y, z) = x + y + z$. A picture of the solid tetrahedron is shown in Figure.

We find the mass $M$ of the tetrahedron using the triple integral

. To do this, we will need to generalize our ideas from Section 4.4 and describe $S$ using three sets of inequalities, one for each variable. In this example, we choose to integrate with respect to $z$ first for the innermost integral. We will first need to consider how to give bounds on the “top” and “bottom” functions for $z$ as a function of $x$ and $y$. In other words, we need to give functions $z=g(x,y)$ and $z=h(x,y)$ such that for any $(x,y)$ point in our region, $g(x,y)$ will give the largest $z$-value we need to consider and $h(x,y)$ will need to give the smallest $z$-value. This description will give us an iterated integral of the form

Note that the inner integral will be considered with $x$ and $y$ held constant and $D$ being the set of $(x,y)$ points over which our three dimensional region of integration sits.

You can see from that the plane containing the points $(6,0,0),(0,3,0),(0,0,2)$ will give us $z=g(x,y)$ and the $xy$-plane will give us $z=h(x,y)$. So we have

as our first iterated integral. We now need to consider $D$, the region of the $xy$-plane over which our region $S$ sits. In this example, $D$ coincides with the triangle in the $xy$ plane with vertices $(0,0),(6,0),(0,3)$, as drawn in .

In order to complete our transformation to iterated integrals, we need to describe $D$ as either horizontally simple or vertically simple. We can see that $D$ is both vertically simple and horizontally simple, so we could use either description. In this example, we chose to describe the region as vertically simple, as suggested by the dashed lines in . We will cut our region into vertical slices for $0\leq x\leq 6$. Furthermore, the lower bound on each slice is $y=0$, so we just need to find the equation of the top boundary of the region. We can find the equation of the line that determines this top boundary as $x + 2 y = 6$. Solving for $y$ gives $y = 3 - \frac{1}{2}x$. Therefore, we can describe the base of the tetrahedron as a vertically simple region using the inequalities

.

We can now combine our vertically simple description of $D$ with our iterated integral above to get

We can restate our work above as a description of the solid $S$ using the inequalities

.

With our description of $S$ in terms of inequalities in hand, we can write an iterated triple integral to find the mass of the tetrahedron by integrating the density function $\delta(x,y,z)=x+y+z$:

. Evaluating the three iterated integrals yields

For the innermost integral of equation, we need bounds on the $z$-coordinate for fixed values of $x$ and $y$. In Figure, you can use the sliders to change the values of $x$ and $y$. When your choices of $x$ and $y$ correspond to points inside the solid, you see a vertical line segment in the plot from the bottom of the solid to the top of the solid over the point $(x,y)$ in the $xy$-plane.

Try several values of $x$ and $y$. Look at how the length of the segment changes in the $z$-direction. In particular, for every $x$ and $y$ pair, the bottom boundary of the solid is the same. Similarly, for every pair of values the top boundary of the solid is the same surface. This allows you to use functions, in terms of $x$ and $y$, that describe the top and bottom boundaries of the solid. These are the $z$-coordinates of the points at the top and bottom of the vertical line segments through the solid shown in the figure.

Complete the compound inequality below to provide lower and upper bounds for the $z$-coordinates of points in $S$ (in terms of $x$ and $y$).

Having established upper and lower bounds for $z$ as a function of a fixed choice of $x$ and $y$, we need to describe the set of points $(x,y)$ in the $xy$-plane such that a vertical line through $(x,y)$ will pass through the solid. Notice that if you choose values of $x$ and $y$ in Figure that does not intersect the solid (e.g., $(x,y)=(2.6,-2.3)$), then the point is shown in red.

You can see from Figure that there will be $x$ and $y$ values from $-3$ to $3$ that will correspond to points in our solid. It is tempting to give the region of the $xy$-plane we need to consider using the inequalities $-3\leq x\leq 3$ and $-3\leq y\leq 3$. Write a couple of sentences to explain why the set of $(x,y)$ points we need to consider is not the square $[-3,3]\times[-3,3]$.

On , draw a plot of $D$, the region of $(x,y)$ points that correspond to points of $S$. We refer to $D$ as the projection of $S$ onto the $xy$-plane.

To complete the task of writing

as an iterated integral, you need to describe the region $D$ from the previous part using inequalities, as with double integrals. Do this using a vertically simple description in order to have your iterated integral fit the form of equation.

Give the inequalities that describe the region $D$ from the previous part as vertically simple.

Write an iterated integral of the form that represents the mass of the cone $S$.

How many different orders of integration could be used for iterated integrals that are equal to the integral in equation?

Set up an iterated integral, integrating first with respect to $z$, then $x$, then $y$ that is equivalent to the integral in equation. Before you write down the integral, think about Figure and draw a plot of the appropriate projection.

Set up an iterated integral, integrating first with respect to $y$, then $z$, then $x$, that is equivalent to the integral in equation. As above, think carefully about the geometry first and draw a plot of the appropriate projection.

Set up an iterated integral, integrating first with respect to $x$, then $y$, then $z$ that is equivalent to the integral in equation.

Set up an iterated triple integral to find the volume of $S$. You do not need to evaluate the integral at this time.

Suppose the density at point $(x,y,z)$ is $\delta(x,y,z)=x^2+1$. Set up, but do not evaluate, the necessary iterated integrals to find the center of mass of $S$.

Let $f(x,y,z) = xy+z^2$. Set up but do not evaluate an iterated triple integral to find the average value of $f$ on $S$.

Use technology to evaluate the iterated triple integrals you wrote in the other three parts of this activity. Write a couple of sentences to explain why the location of the center of mass makes sense.